# Project Lyra: A Mission to 1I/’Oumuamua without Solar Oberth Manoeuvre

Adam Hibberd  Initiative for Interstellar Studies (i4is) 27/29 South Lambeth
Road London, SW8 1SZ United Kingdom  Andreas M. Hein  Initiative for
Interstellar Studies (i4is) 27/29 South Lambeth Road London, SW8 1SZ United
Kingdom  T. Marshall Eubanks  Space Initiatives Inc, Newport, VA 24128, USA
Robert G. Kennedy III  Initiative for Interstellar Studies (i4is) 27/29 South
Lambeth Road London, SW8 1SZ United Kingdom

###### Abstract

To settle the question of the nature of the interstellar object 1I/’Oumuamua
requires in-situ observations via a spacecraft, as the object is already out
of range of existing telescopes. Most previous proposals for reaching
1I/’Oumuamua using near-term technologies are based on the Solar Oberth
Manoeuvre (SOM), as trajectories without the SOM are generally significantly
inferior in terms of lower mission duration and higher total velocity
requirement. While the SOM allows huge velocity gains, it is also technically
challenging and thereby increases programmatic and mission-related risks. In
this paper, we identify an alternative route to the interstellar object
1I/’Oumuamua, based on a launch in 2028, which does not require a SOM but has
a similar performance as missions with a SOM. It instead employs a Jupiter
Oberth Manoeuvre (JOM) with a total time of flight of around 26 years or so.
The efficacy of this trajectory is a result of it significantly reducing the
$\Delta$V to Jupiter by exploiting the VEEGA sequence. The total $\Delta$V of
the trajectory is 15.8 $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ and the
corresponding payload mass is 115 $\mathrm{k}\mathrm{g}$ for a SLS Block 1B or
241 $\mathrm{k}\mathrm{g}$ for a Block 2. A further advantage of the JOM is
that the arrival speed relative to 1I/’Oumuamua is approximately 18
$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$, much lower than the equivalent for
the SOM of around 30 $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$.

tablenum SIXtablenum

##  1 Introduction

The celestial body designated 1I/’Oumuamua is the first object to be
discovered and identified as an interstellar object, stimulating much
interest, debate and speculation from the scientific community (Bannister et
al., 2019). Theories to explain the nature of 1I/’Oumuamua have included a
fractal dust aggregate (Flekkøy et al., 2019), a hydrogen iceberg (Seligman &
Laughlin, 2020), a nitrogen iceberg (Jackson & Desch, 2021; Desch & Jackson,
2021), an alien solar sail (Bialy & Loeb, 2018), fragments of a tidally
disrupted planet (Raymond et al., 2018) and so on. All explanations have one
feature in common - they are extraordinary. Given the enigmatic nature of 1I
and to settle this matter, additional data would be vital. However, due to the
large distance 1I has already traveled away, it is no longer possible to
obtain data via telescopes. It has been argued that therefore, we should
rather wait for the next similar object to come within range of telescopes.

However, the discovery of the second interstellar object 2I/Borisov in 2019
(Jewitt & Luu, 2019) confirmed that there is at least one interstellar object
which resembles minor bodies found in the Solar System. This makes 1I even
more of an oddity and it is unclear what the likelihood of encountering a
similar object again is, although various estimates have been given in the
literature (Portegies Zwart et al., 2018; Bannister et al., 2019; Desch &
Jackson, 2021). Hence, waiting for the next object similar to 1I to enter our
Solar System might be fruitless. An alternative strategy would be to send a
spacecraft to 1I. Despite initial statements against the feasibility of this
option (Bialy & Loeb, 2018), a series of publications has argued that such a
mission would be feasible with near-term technologies (Hein et al., 2019;
Seligman & Laughlin, 2018; Hibberd & Hein, 2021a, b; Hibberd et al., 2020).

As a contribution to this effort, previous research has been conducted into
spacecraft missions to 1I/’Oumuamua (Hein et al., 2019; Seligman & Laughlin,
2018; Hibberd & Hein, 2021a, b; Hibberd et al., 2020), and the possible
scientific return from such a venture make this an unmissable opportunity
(Hein et al., 2022; Eubanks et al., 2020). Given the limitations of present or
near-term technology, these studies have generally identified a Solar Oberth
Manoeuvre (SOM) as a requirement for viable spacecraft missions. As an
exception, Hibberd et al. (2020) have identified several trajectories without
a SOM. However, these trajectories seem to be generally inferior in terms of
longer mission duration and higher required $\Delta$V, compared to missions
with a SOM.

While the advantages of the SOM have been demonstrated, the recent
Interstellar Probe endeavour has highlighted the multiple drawbacks of such a
manoeuvre (McNutt Jr. et al., 2021) and in its place has proposed either 1) a
passive gravitational assist of Jupiter, or 2) a powered GA of Jupiter (a
Jupiter Oberth), the SOM being the least preferred.

While the arguments against the SOM can be debated, investigating the prospect
of a mission to 1I without the SOM but comparable performance seems to merit
consideration, as it would certainly alleviate some of the technological
challenges of the SOM such as the development of a scaled up Parker Solar
Probe heat shield and a precise thrust orientation during the Manoeuvre.
Without the SOM, the mission would much more resemble existing interplanetary
missions.

In this context, here we present an alternative solution, a mission to
’Oumuamua which has relatively low $\Delta$V, yet does not conduct a SOM to
accelerate the craft. The mission is abbreviated as V-E-DSM-E-J , adopting the
convention that the home planet (E) and target body (1I) are implicitly
assumed. Usually in the context of a mission to Jupiter, this is alternatively
referred to as VEEGA.

##  2 Method

As in previous Project Lyra studies, The ’Optimum Interplanetary Trajectory
Software’ (OITS) (Hibberd, 2017) was employed to research missions to
’Oumuamua. It adopts the following assumptions:

  1. 1.

a patched conic for every planet encountered

  2. 2.

the thrust is impulsive resulting in discrete applications of $\Delta$V along
the trajectory

  3. 3.

the $\Delta$V is applied at the point of periapsis w.r.t. every planet
encountered

  4. 4.

this $\Delta$V is applied tangential to the motion of the spacecraft at every
planet encountered

As an alternative instead of a planet, an ’Intermediate Point’ may be chosen,
a point whose heliocentric longitude and latitude are optimised by OITS but
whose sun distance is user-specified. This can be selected to model a Deep
Space Manoeuvre (DSM).

Two Non-Linear Programming (NLP) optimizers were used for this study, firstly
NOMAD (Le Digabel, 2011) for a general solution and to fine-tune the results
MIDACO (Schlueter et al., 2009; Schlueter & Gerdts, 2010; Schlueter et al.,
2013).

##  3 Results

Table 1: Mission V-E-DSM-E-J (VEEGA) with 22 Years Flight Duration Constraint Num | Planet | Time | Arrival speed | Departure speed |  $\Delta$V | Cumulative $\Delta$V | Periapsis
---|---|---|---|---|---|---|---
|  |  | $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ | $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ | $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ | $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ | $\mathrm{k}\mathrm{m}$
1 | Earth | 2028 FEB 04 | 0.0000 | 4.1973 | 4.1973 | 4.1973 | N/A
2 | Venus | 2028 JUN 23 | 6.7660 | 5.0616 | 0.8536 | 5.0509 | 200.0
3 | Earth | 2029 FEB 27 | 7.5338 | 7.5338 | 0.0000 | 5.0510 | 2900.2
4 | DSM @ 2.2 au | 2030 JAN 24 | 15.8493 | 15.4643 | 0.4184 | 5.4693 | N/A
5 | Earth | 2031 FEB 09 | 9.8911 | 12.9506 | 2.1978 | 7.6671 | 200.0
6 | Jupiter | 2032 APR 17 | 16.8277 | 41.1420 | 10.5122 | 18.1793 | 200.0
7 | ’Oumuamua | 2050 JAN 29 | 22.4983 | 22.4983 | 0.0000 | 18.1793 | N/A
Table 2: Mission V-E-DSM-E-J (VEEGA) with 26 Years Flight Duration Constraint Num | Planet | Time | Arrival speed | Departure speed |  $\Delta$V | Cumulative $\Delta$V | Periapsis
---|---|---|---|---|---|---|---
|  |  | $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ | $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ | $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ | $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ | $\mathrm{k}\mathrm{m}$
1 | Earth | 2028 MAR 07 | 0.0000 | 3.4550 | 3.4550 | 3.4550 | N/A
2 | Venus | 2028 JUL 06 | 5.6526 | 5.6511 | 0.0070 | 3.4557 | 200.0
3 | Earth | 2029 MAR 13 | 8.0443 | 6.2791 | 0.9708 | 4.4265 | 393.6
4 | DSM @ 2.2 au | 2030 FEB 12 | 15.9254 | 15.4781 | 0.4518 | 4.8783 | N/A
5 | Earth | 2031 FEB 15 | 9.0627 | 13.2924 | 3.0000 | 7.8783 | 200.0
6 | Jupiter | 2032 MAR 22 | 18.5791 | 37.0710 | 7.9588 | 15.8371 | 4464.5
7 | ’Oumuamua | 2054 MAR 01 | 18.1348 | 18.1348 | 0.0000 | 15.8371 | N/A
Table 3: Mass Breakdown for Jupiter Oberth, assuming a total Jupiter $\Delta$V of 8.0 $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ |  |  | Stage2 |
---|---|---|---|---
|  | STAR 75 | STAR 63F | STAR 48B
| STAR 75 | 342 kg | 406 kg | 378 kg
|  | 16478 kg | 13064 kg | 10580 kg
Stage 1 | STAR 63F | 169 kg | 239 kg | 241 kg
|  | 12827 kg | 9419 kg | 6966 kg
| STAR 48B | 26 kg | 94 kg | 115 kg
|  | 10228 kg | 6818 kg | 4383 kg
Table 4: Launch Vehicle Data for $C_{3}=12$\mathrm{k}\mathrm{m}^{2}\,\mathrm{s}^{-2}$$ Launch Vehicle | Payload Mass to $C_{3}=12$\mathrm{k}\mathrm{m}^{2}\,\mathrm{s}^{-2}$$ (kg) | Remaining Mass for JOM (kg)
---|---|---
SLS Block 1B | 32000 | 6468
SLS Block 2B | 36000 | 7276
Falcon Heavy Expendable | 11880 | 2401

The solution trajectory is shown in Figure 1 (coinciding with the 2028 VEEGA
trajectory in Petropoulos et al. (2000)) and the numerical information is
provided in Table 1 . This scenario amounts to the same VEEGA sequence to
Jupiter adopted for instance by the Galileo spacecraft, where it was shown to
be the globally optimal trajectory for the mission (Strange & Longuski, 2002).
Furthermore Zubko et al. (2021) also exploit this sequence in their analysis
of missions to Sedna. Note the mission duration constraint of 22 years (as
also imposed in Hibberd et al. (2020)) results in a $\Delta$V of 10.5
$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ delivered at Jupiter (row 6). This is
rather larger than is easily reached by two staged solid rocket motors (SRMs),
so in order to reduce this $\Delta$V, the flight duration restriction may be
relaxed.

Figure 1: Trajectory, V-E-DSM-E-J (VEEGA) Figure 2:  $\Delta$V vs Flight
Duration, V-E-DSM-E-J (VEEGA) Figure 3: Trajectory, E-J-DSM-J

Figure 2 is a graph revealing the dependency of overall $\Delta$V and the
Jupiter $\Delta$V on mission duration constraint for the V-E-DSM-E-J, VEEGA
scenario. On the secondary vertical axis we have the magnitude of $C_{3}$
(equivalent to the square of the hyperbolic excess speed at Earth departure -
row 1 of Table 1).

From this Figure we adopt the 26 year mission duration as it approximately
coincides with the minimum $C_{3}$ at launch. In order to ensure the in-flight
$\Delta$V allows room for a significant payload to be delivered to
1I/’Oumuamua, an extra constraint was applied to the $\Delta$V at the second
Earth gravitational assist. Thus refer to Table 2 wherein we see a constraint
of 3.0 $\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ is imposed at this encounter
(row 5). An animation of this trajectory can be found at Adam’s Space Research
(2022).

Addressing the Jupiter Oberth Manoeuvre (JOM- Row 6 of Table 2), we observe to
deliver such a hefty kick in velocity, 8.0
$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$, we need as a minimum 2 staged SRMs.
We assume some combination of STAR 75, STAR 63F and STAR 48B is exploited, the
situation is summarised in Table 3. All the possible permutations of these
SRMs are accounted for and we find that the payload mass for any combination
is positive.

Preceding the JOM, there is a total of 4.4
$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$ of $\Delta$V applied en route. Let us
assume that all of this $\Delta$V is delivered by a throttleable Liquid
Propellant Engine (assuming $MMH/N_{2}O_{4}$ fuel, $I_{sp}$=341 $\mathrm{s}$,
$p=0.1$). In addition we find that for the 26 year duration mission, there is
a $C_{3}$ value at Earth of $3.455^{2}=11.94$
$\mathrm{k}\mathrm{m}^{2}\mathrm{s}^{-2}$. Table 4 provides the payload mass
capability of 3 launch vehicles to a $C_{3}$=12
$\mathrm{k}\mathrm{m}^{2}\mathrm{s}^{-2}$ escape orbit, the Space Launch
System (SLS) Block 1B, the SLS Block 2 and a Falcon Heavy Expendable.

Referencing Tables 3 and 4, we find that the Block 1B with two staged STAR 48B
SRMs can deliver 115 $\mathrm{k}\mathrm{g}$ to ’Oumuamua, whereas the Block 2
is capable of 241 kg, with a STAR 63F followed by a STAR 48B. The Falcon Heavy
Expendable, however, is unable to achieve enough payload to deliver the
necessary JOM $\Delta$V, even assuming 2 STAR 48B SRMs.

At this juncture as a side note, we reiterate a previously discovered
trajectory in Hibberd et al. (2020). It is depicted in Figure 3 and can be
abbreviated as E-J-DSM-J. The DSM is placed at 5.2 $\mathrm{a}\mathrm{u}$ from
the sun and at a low (negative) heliocentric latitude. The relevance is that
it is also an option which does not require a SOM, and in fact it exploits 2
JOMs in succession, separated by half a Jupiter cycle, i.e. approximately 6
years. The launch would have to be be in 2023, which is unfortunately too soon
to organise a mission using this scenario.

##  4 Discussion

We have demonstrated that at least one trajectory for a mission to 1I exists
without a SOM. A mission with such a trajectory has multiple advantages:

  * •

The development of a scaled up version of the Parker Solar Probe heat shield
can be omitted, thereby decreasing development risk.

  * •

The mass of the heat shield is thereby removed from the mass budget, freeing
up mass that can be dedicated to other spacecraft subsystems.

  * •

The mission risk of a precise steering of the thrust vector during the Oberth
Manoeuvre is eliminated. The velocity vector at the end of the Oberth
Manoeuvre is very sensitive to the thrust vector applied to the spacecraft
during the Manoeuvre. Even small errors can lead to a large deviation of the
spacecraft from its target trajectory.

  * •

The mission much closer resembles the existing heritage of interplanetary
missions and decreases its implementation risk.

  * •

The arrival velocity of the spacecraft w.r.t. 1I is lower (18
$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$) than that for the SOM (30
$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$).

There are also disadvantages with missions to 1I without an SOM:

  * •

Due to the large number of flybys, launch opportunities are much rarer than
for missions with an SOM.

  * •

The mission duration is longer than for missions with an SOM: about 25 years
in the present case versus mission durations of $<$ 20 years.

##  5 Conclusion

In this paper, we presented a mission and trajectory design to 1I/’Oumuamua
without a Solar Oberth Manoeuvre but comparable performance. Our results
indicate that with a combination of Venus, Earth, and Jupiter flybys,
including a Deep Space Manoeuvre, a mission to 1I can be staged in 2028 with
reaching 1I in 2054. The total $\Delta$V of the mission would be 15.8
$\mathrm{k}\mathrm{m}\,\mathrm{s}^{-1}$. The absence of a Solar Oberth
Manoeuvre significantly reduces the development risks of such a mission,
mainly due to elimination of the heat shield. The freed up mass can be
allocated to other spacecraft subsystems. Also, the specific mission risks of
a Solar Oberth Manoeuvre, such as the precise burn and velocity vector
orientation at the end of the Manoeuvre are alleviated. However, a clear
disadvantage is that launch opportunities for such a mission are much rarer
and therefore reduce the flexibility. Also, the complexity of the mission with
its multiple flybys is increased. This may, nevertheless, not be a significant
drawback, as current interplanetary missions face the same issues. Finally,
our results rather strengthen the feasibility of a mission to 1I, as we
demonstrated that such a mission may resemble existing interplanetary missions
and not rely on significant new technology developments or unprecedented space
Manoeuvres.

## References

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Separation of Variables, Quasi-Trigonometric $r$-Matrices and Generalized
Gaudin Models

Separation of Variables, Quasi-Trigonometric
$\boldsymbol{r}$-Matrices and Generalized Gaudin Models

Taras SKRYPNYK

T. Skrypnyk

Bogolyubov Institute for Theoretical Physics, 14-b Metrolohichna Str., Kyiv,
03680, Ukraine<EMAIL_ADDRESS>
Received March 29, 2021, in final form July 07, 2021; Published online July
18, 2021

We construct two new one-parametric families of separated variables for the
classical Lax-integrable Hamiltonian systems governed by a one-parametric
family of non-skew-symmetric, non-dynamical
$\mathfrak{gl}(2)\otimes\mathfrak{gl}(2)$-valued quasi-trigonometric classical
$r$-matrices. We show that for all but one classical $r$-matrices in the
considered one-parametric families the corresponding curves of separation
differ from the standard spectral curve of the initial Lax matrix. The
proposed scheme is illustrated by an example of separation of variables for
$N=2$ quasi-trigonometric Gaudin models in an external magnetic field.

integrable systems; separation of variables; classical $r$-matrices

14H70; 17B80; 37J35

##  1 Introduction

The problem of the separation of variables is one of the most studied, but yet
the most difficult and not completely resolved problems in the theory of
integrable systems. The separated variables are important because they give
one a possibility to integrate classical equations of motion [9]. They are
also used to solve exactly quantum integrable systems [19]. In this context it
is worth to mention, for example, the recent results in quantum separation of
variables (SoV) for the higher-rank models [7, 15, 16].

The problem of construction of separated variables can be divided into three
parts:

  1. 1.

Construction of a set of canonical coordinates on the phase space.

  2. 2.

Proof that the constructed coordinates satisfy the so-called equations of
separation.

  3. 3.

Proof that the constructed system of canonical coordinates is complete.

The important class of integrable models are the models admitting Lax
representation. For such the models one can much advance in the solution of
the problem of separation of variables [19]. In the classical case the
approach of [19] – the roots of which go back to the previous papers [1, 3, 8,
12, 18, 30] – permits to construct a set of variables that belong to a
spectral curve of the Lax matrix, playing the role of equation of separation,
i.e., the approach of [19] – the so-called “magic recipe” – automatically
guarantees the validity of item two.

Unfortunately “the magic recipe” does not guarantee that the constructed
variables belonging to the spectral curve of the Lax matrix are the canonical
variables indeed. Moreover, it often happens that the number of variables
produced by the method is not equal to the dimension of the corresponding
phase space, i.e., item three is not guaranteed even if both of the items one
and two are fulfilled. Although in some cases the problem can be resolved by
certain tricks, e.g., by complementing of the set of the obtained canonical
variables by a linear integrals playing the role of the additional momenta of
separation and by finding of the corresponding canonically conjugated
variables [1, 2, 18] but, unfortunately, it is not always the case.

That is why we propose to re-consider a scheme of [19], making its starting
point a construction of the complete set of canonical coordinates (items 1, 3)
and only after that start to check item 2, i.e., to check what equations of
separation the constructed canonical coordinates satisfy. This changing of
accents permits to obtain separated variables for which the equations of
separation do not coincide with a spectral curve of the initial Lax matrix
[29].

In order to find a complete set of the canonical coordinates we use, similar
to [19], the method of separating functions $B(u)$ and $A(u)$, such that zeros
of the first function generate the Poisson-commuting coordinates and the
values of the second function in these zeros generate the canonically
conjugated momenta: $B(x_{i})=0$, $p_{i}=A(x_{i})$. The most important in this
method is a Poisson algebra to be satisfied by the functions $B(u)$ and $A(u)$
[10]:

$\displaystyle\\{B(u),B(v)\\}=b(u,v)B(u)-b(v,u)B(v),$ (1.1a)
$\displaystyle\\{A(u),B(v)\\}=\alpha(u,v)B(u)-\beta(u,v)B(v),$ (1.1b)
$\displaystyle\\{A(u),A(v)\\}=a(u,v)B(u)-a(v,u)B(v),$ (1.1c)

for some functions $a(u,v)$, $b(u,v)$, $\alpha(u,v)$, $\beta(u,v)$ such that
the following limit holds true

$\lim\limits_{u\rightarrow
v}(\alpha(u,v)B(u)-\beta(u,v)B(v))=\partial_{v}B(v)+\gamma(v)B(v).$

The corresponding restrictions imposed on $B(u)$ and $A(u)$ permit their
explicit construction.

In the present paper we consider the examples of Lax-integrable models with
$\mathfrak{gl}(2)$-valued Lax matrices $L(u)=\sum_{i,j=1}^{2}L_{ij}(u)X_{ij}$
possessing the tensor Lie–Poisson bracket

$\\{L(u)\otimes 1,1\otimes L(v)\\}=[r_{12}(u,v),L(u)\otimes
1]-[r_{21}(v,u),1\otimes L(v)],$

governed by $\mathfrak{gl}(2)\otimes\mathfrak{gl}(2)$-valued non-skew-
symmetric “quasi-trigonometric” classical $r$-matrix

$\displaystyle
r(u,v)=\left(\frac{1}{2}\frac{(u+v)}{(v-u)}+c_{1}\right)X_{11}\otimes
X_{11}+\left(\frac{1}{2}\frac{(u+v)}{(v-u)}+c_{2}\right)X_{22}\otimes X_{22}$
$\displaystyle\hphantom{r(u,v)=}{}+\frac{v}{(v-u)}X_{12}\otimes
X_{21}+\frac{u}{(v-u)}X_{21}\otimes X_{12},$ (1.2)

where the parameters $c_{i}$ are arbitrary and $\\{X_{ij}\mid i,j=1,2\\}$ is a
standard matrix basis of $\mathfrak{gl}(2)$, i.e., the matrices $X_{ij}$ have
the following matrix elements:
$(X_{ij})_{\alpha\beta}=\delta_{i\alpha}\delta_{j\beta}$.

Although the mentioned above trick with the complement of the standard
Sklyanin variables is valid in the considered cases (see [14] for the standard
trigonometric case), i.e., it helps to produce a complete set of variables of
separation, in the present paper we have succeeded in a construction of two
new one-parametric families of separating functions for the $r$-matrix (1.2)
in the subcase $c_{2}=-c_{1}$, which are complete at once without any
additional tricks.

The first family has the following explicit form

$\displaystyle
B(u)=L_{12}(u)+\frac{k}{u}(L_{11}(u)-L_{22}(u))-\frac{k^{2}}{u^{2}}L_{21}(u),$
(1.3a) $\displaystyle
A(u)=L_{22}(u)+\frac{k}{u}L_{21}(u)+\left(c_{1}+\frac{1}{2}\right)M,$ (1.3b)

where the parameter $k$ is arbitrary and $M$ is a special “geometric” integral
[26] possessing the following Poisson brackets with the matrix elements of the
Lax matrix:

$\\{M,L_{ij}(u)\\}=\operatorname{sign}(j-i)(1-\delta_{ij})L_{ij}(u).$

The second family is organized in a similar way. In the case $k\neq 0$ both
the families produce the complete set of canonical coordinates. The important
property of the both sets of separated coordinates is that they satisfy the
spectral curve equation modified with the help of the geometric integral $M$.
For the first family of separated variables the equations of separation have
the following explicit form

$\det\left(L(x_{j})-\left(x_{j}p_{j}-\left(c_{1}+\frac{1}{2}\right)M\right){\rm
Id}\right)=0.$ (1.4)

They coincides with a spectral curve of the initial Lax matrix only in the
special case $c_{1}=-\frac{1}{2}$.

The function $B(u)$ given by (1.3a) and the function $A(u)$ given by (1.3b)
are obtained – modulo the integral $M$ – using the gauge transformation $K(u)$
of the initial Lax matrix (see Remarks 4.5 and 4.10). Due to the fact that the
gauge-transformed $r$-matrix $r^{K}(u,v)=K^{-1}(u)\otimes
K^{-1}(v)r(u,v)K(u)\otimes K(v)$ does not satisfy the conditions of [10] (see
also [28]), we have inserted the integral $M$ into the momenta-generating
function $A(u)$ in order to achieve the needed structure (1.1) of Poisson
algebra of separating functions.

Hence, the main difference of our approach with the standard one is in the
form of the momenta-generating function (1.3b) and of the equations of
separation (1.4). It evidently leads to the SoV non-equivalent with the
standard one. Our SoV is naturally characterized by another form of the Abel-
type equations and of the reconstruction formulae. Besides the difference in
the form of separation curves leads to the profound mathematical difference of
our SoV with the standard SoV based on the usual spectral curve. In
particular, the proposed SoV is not bi-Hamiltonian one (see [11]), since – as
the result of the $M$-shift – the integrals of the corresponding models do not
enter into the equations of separation (1.4) in the form of the Casimirs of
the corresponding Poisson pencil.

In order to be more concrete we consider a class of examples of the Lax
matrices possessing the poles of the first order in the points
$\nu_{1},\nu_{2},\dots,\nu_{N}$, i.e., the Lax matrices of the generalized
Gaudin models [20, 21, 22]. We devote special attention to the case $N=2$. The
corresponding Poisson-commuting Gaudin-type Hamiltonians in an external
magnetic field are written as follows

$\displaystyle
H_{1}=\left(\frac{1}{2}\frac{(\nu_{1}+\nu_{2})}{(\nu_{2}-\nu_{1})}+c_{1}\right)T_{11}S_{11}+\left(\frac{1}{2}\frac{(\nu_{1}+\nu_{2})}{(\nu_{2}-\nu_{1})}+c_{2}\right)T_{22}S_{22}$
$\displaystyle\hphantom{H_{1}=}{}+\frac{\nu_{1}T_{12}S_{21}}{(\nu_{2}-\nu_{1})}+\frac{\nu_{2}T_{21}S_{12}}{(\nu_{2}-\nu_{1})}+c_{1}S^{2}_{11}+c_{2}S^{2}_{22}+2c_{11}S_{11}+2c_{22}S_{22},$
$\displaystyle
H_{2}=\left(\frac{1}{2}\frac{(\nu_{1}+\nu_{2})}{(\nu_{1}-\nu_{2})}+c_{1}\right)S_{11}T_{11}+\left(\frac{1}{2}\frac{(\nu_{1}+\nu_{2})}{(\nu_{1}-\nu_{2})}+c_{2}\right)S_{22}T_{22}$
$\displaystyle\hphantom{H_{2}=}{}+\frac{\nu_{2}S_{12}T_{21}}{(\nu_{1}-\nu_{2})}+\frac{\nu_{1}S_{21}T_{12}}{(\nu_{1}-\nu_{2})}+c_{1}T^{2}_{11}+c_{2}T^{2}_{22}+2c_{11}T_{11}+2c_{22}T_{22},$

where $c_{ii}$ are the components of an external magnetic field and

$\\{{S}_{ij},{S}_{kl}\\}=\delta_{kj}{S}_{il}-\delta_{il}{S}_{kj},\qquad\\{{T}_{ij},{T}_{kl}\\}=\delta_{kj}{T}_{il}-\delta_{il}{T}_{kj},\qquad\\{T_{ij},S_{kl}\\}=0.$

For the considered $N=2$ Gaudin models we explicitly write coordinates and
momenta of separation, the reconstruction formulae and the Abel-type equations
corresponding to the both constructed one-parametric families of separating
functions.

We remark, that our previous result [29] on separation of variables in
trigonometric models is recovered, in the case $c_{1}=c_{2}=0$ and for certain
fixed value of the parameter $k$, from our first family of separated
coordinates after suitable re-parametrization and gauge transformation.

The structure of the present article is the following: in Section 2 we remind
main facts about separation of variables and about the method of separating
functions, in Section 3 we recall the general theory of the classical
$r$-matrices and generalized Gaudin models and specialize this theory for the
case of quasi-trigonometric $r$-matrices. In Section 4 we construct two
families of separating functions for any integrable system governed by the
quasi-trigonometric $r$-matrix. In Section 5 we perform variable separation
for the $N=2$ quasi-trigonometric Gaudin models. In Section 6 we briefly
conclude and discuss the on-going problems. At last in Appendices A and B we
give the explicit form of the reconstruction formulae for the both cases of
SoV discovered in the present paper and for $N=2$ quasi-trigonometric Gaudin
models.

##  2 Separation of variables: general scheme

###  2.1 Definitions

Let us recall the definition of Liouville integrability and separation of
variables in the theory of Hamiltonian systems [19]. An integrable Hamiltonian
system with $D$ degrees of freedom is determined on a $2D$-dimensional
symplectic manifold $\mathcal{M}$ – symplectic leaf in the Poisson manyfold
$(\mathcal{P},\\{\ ,\ \\})$ – by $D$ independent first integrals $I_{j}$
commuting with respect to the Poisson bracket

$\\{I_{i},I_{j}\\}=0,\qquad i,j=1,\dots,D.$

For the Hamiltonian $H$ of the system may be taken any first integral $I_{j}$.

To find separated variables means to find (at least locally) a set of
coordinates $x_{i}$, $p_{j}$, $i,j=1,\dots,D$ such that there exist $D$
relations – equations of separation

$\Phi_{i}(x_{i},p_{i},I_{1},\dots,I_{D})=0,\qquad i=1,\dots,D,$ (2.1)

where the coordinates $x_{i}$, $p_{j}$, $i,j=1,\dots,D$ are canonical, i.e.,

$\\{x_{i},p_{j}\\}=\delta_{ij},\qquad\\{x_{i},x_{j}\\}=0,\qquad\\{p_{i},p_{j}\\}=0,\qquad\forall\,i,j=1,\dots,D.$

The separated variables provide a way to a construction of the action-angle
coordinates from the Liouville theorem and a way to explicit integration of
the equations of motion.

Unfortunately, in the general case no algorithm is known to construct a set of
separated variables for any given integrable system. One of the possible
methods of their construction is the so-called method of separating functions
permitting one to construct a set of canonical coordinates.

###  2.2 Separating functions and canonical coordinates

Let us remind a method of construction of canonical coordinates using
separating functions. Generally speaking this method can be considered
independently of separation of variables. That is why in this subsection we do
not assume any special properties of the Poisson manyfold $\mathcal{P}$ or
Poisson structure $\\{\ ,\ \\}$. Neither we assume integrability or existence
of the Lax representation.

Let $B(u)$ and $A(u)$ be some functions of the dynamical variables and an
auxiliary parameter $u$, which is constant with respect to the bracket $\\{\
,\ \\}$. Let the points $x_{i}$, $i=1,\dots,P$ be zeros of the function $B(u)$
and $p_{i}$, $i=1,\dots,P$ be the values of $A(u)$ in these points, i.e.,

$B(x_{i})=0,\qquad p_{i}=A(x_{i}).$

We wish to construct Poisson brackets among these new coordinates using the
Poisson brackets between $B(u)$ and $A(u)$. The following proposition holds
true [10].

######  Proposition 2.1.

Let $B(x_{i})=0$, $p_{j}=A(x_{j})$. Then

$\displaystyle(i)\ $
$\displaystyle\\{x_{i},x_{j}\\}=\left(\frac{\\{B(u),B(v)\\}}{\partial_{u}B(u)\partial_{v}B(v)}\right)\bigg{|}_{u=x_{i},\,v=x_{j}},$
$\displaystyle(ii)\ $
$\displaystyle\\{x_{j},p_{i}\\}=\left(\frac{\\{A(u),B(v)\\}}{\partial_{v}B(v)}\right)\bigg{|}_{u=x_{i},\,v=x_{j}}+\\{x_{i},x_{j}\\}(\partial_{u}A(u))|_{u=x_{i}},$
$\displaystyle(iii)\ \ $
$\displaystyle\\{p_{i},p_{j}\\}=\big{(}\\{A(u),A(v)\\}\big{)}\big{|}_{u=x_{i},\,v=x_{j}}+\\{p_{i},x_{j}\\}(\partial_{v}A(v))|_{v=x_{j}}+\\{x_{i},p_{j}\\}(\partial_{u}A(u))|_{u=x_{i}}$
$\displaystyle\quad{}-\\{x_{i},x_{j}\\}(\partial_{u}A(u)\partial_{v}A(v))|_{u=x_{i},\,v=x_{j}},$

where $i\neq j$.

###### Proof.

The equalities (i)–(iii) are obtained by the decomposition of $B(u)$, $A(u)$,
$B(v)$, $A(v)$ in Taylor power series in the neighborhood of the points
$u=x_{i}$, $v=x_{j}$ in the expressions $\\{B(u),B(v)\\}$, $\\{A(u),B(v)\\}$,
$\\{A(u),A(v)\\}$ and by considering the limits $u\rightarrow x_{i}$,
$v\rightarrow x_{j}$ after the calculation of the Poisson brackets. ∎

Now we are ready to formulate the following important lemma [10].

######  Lemma 2.2.

Let the coordinates $x_{i}$ and $p_{j}$, $i,j=1,\dots,P$ be defined as above.
Let the functions $A(u)$, $B(u)$ satisfy the following Poisson algebra

$\displaystyle\\{B(u),B(v)\\}=b(u,v)B(u)-b(v,u)B(v),$ (2.2a)
$\displaystyle\\{A(u),B(v)\\}=\alpha(u,v)B(u)-\beta(u,v)B(v),$ (2.2b)
$\displaystyle\\{A(u),A(v)\\}=a(u,v)B(u)-a(v,u)B(v).$ (2.2c)

Then the Poisson bracket among the functions $x_{i}$ and $p_{j}$,
$\forall\,i,j=1,\dots,P$, $i\neq j$ are trivial

$\displaystyle\\{x_{i},x_{j}\\}=0,\qquad\forall\,i,j=1,\dots,P,$
$\displaystyle\\{x_{j},p_{i}\\}=0,\qquad\text{if}\quad i\neq j,$
$\displaystyle\\{p_{i},p_{j}\\}=0,\qquad\forall,i,j\in=1,\dots,P.$

If, moreover holds also the condition

$\lim\limits_{u\rightarrow
v}(\alpha(u,v)B(u)-\beta(u,v)B(v))=\partial_{v}B(v)+\gamma(v)B(v)$ (2.3)

then the corresponding Poisson brackets are canonical, i.e.,
$\\{x_{i},p_{i}\\}=1$, $\forall\,i=1,\dots,P$.

######  Remark 2.3.

Observe, that the method of the separating functions $A(u)$ and $B(u)$ does
not, generally speaking, guarantee that the number $P$ of the constructed
canonical variables is equal to $D$, i.e., to half of the dimension of the
generic symplectic leaf. Neither it guarantees that the constructed canonical
coordinates satisfy the equations of separation (2.1) for some integrable
Hamiltonian system defined by the Poisson-commuting Hamiltonians
$\\{I_{i},\,i=1,\dots,D\\}$. Nevertheless it is often the case and it is
necessary only to find the explicit form of the corresponding functions
$\Phi_{i}(x_{i},p_{i},I_{1},\dots,I_{D})$. In the next sections we will
illustrate this by a class of new examples.

##  3 Classical $\boldsymbol{r}$-matrices and generalized Gaudin models

###  3.1 Definition and notations

Let $\mathfrak{g}=\mathfrak{gl}(2)$ be the Lie algebra of the general linear
group over the field of complex numbers. Let $X_{ij}$, $i,j=1,2$ be a standard
basis in $\mathfrak{gl}(2)$ with the commutation relations

$[X_{ij},X_{kl}]=\delta_{kj}X_{il}-\delta_{il}X_{kj}.$

######  Definition 3.1.

A function of two complex variables $r(u_{1},u_{2})$ with values in the tensor
square of the algebra $\mathfrak{g}=\mathfrak{gl}(2)$ is called a classical
$r$-matrix if it satisfies the following generalized classical Yang–Baxter
equation

$[r_{12}(u_{1},u_{2}),r_{13}(u_{1},u_{3})]=[r_{23}(u_{2},u_{3}),r_{12}(u_{1},u_{2})]-[r_{32}(u_{3},u_{2}),r_{13}(u_{1},u_{3})],$
(3.1)

where

$\displaystyle
r_{12}(u_{1},u_{2})\equiv\sum\limits_{i,j,k,l=1}^{2}r^{ij,kl}(u_{1},u_{2})X_{ij}\otimes
X_{kl}\otimes 1,$ $\displaystyle
r_{13}(u_{1},u_{3})\equiv\sum\limits_{i,j,k,l=1}^{2}r^{ij,kl}(u_{1},u_{3})X_{ij}\otimes
1\otimes X_{kl},$

etc. and $r^{ij,kl}(u,v)$ are matrix elements of the $r$-matrix $r(u,v)$.

######  Remark 3.2.

The definition (3.1) holds true also for other semisimple (reductive) Lie
algebras $\mathfrak{g}$. It has appeared in the different forms in the papers
[4, 5, 13]. In the case of skew-symmetric $r$-matrices, i.e., when
$r_{12}(u_{1},u_{2})=-r_{21}(u_{2},u_{1})$ the generalized classical
Yang–Baxter equation reduces to the usual classical Yang–Baxter equation [6,
17].

In the present paper we are interested only in the meromorphic $r$-matrices
that possess the following decomposition

$r(u_{1},u_{2})=\frac{\Omega}{u_{1}-u_{2}}+r_{0}(u_{1},u_{2}),$ (3.2)

where $r_{0}(u_{1},u_{2})$ is a holomorphic
$\mathfrak{gl}(2)\otimes\mathfrak{gl}(2)$-valued function and
$\Omega=\sum_{i,j=1}^{2}X_{ij}\otimes X_{ji}$.

We will need also the following definitions:

######  Definition 3.3.

The classical $r$-matrix is called
$\mathfrak{g}_{0}\subset\mathfrak{g}$-invariant if

$[r_{12}(u_{1},u_{2}),X\otimes 1+1\otimes
X]=0,\qquad\forall\,X\in\mathfrak{g}_{0}.$ (3.3)

######  Definition 3.4.

A $\mathfrak{gl}(2)$-valued function $c(u)=\sum_{i,j=1}^{2}c_{ij}(u)X_{ij}$ of
one complex variable is called a generalized shift element if it satisfies the
following equation

$[r_{12}(u_{1},u_{2}),c(u_{1})\otimes 1]-[r_{21}(u_{2},u_{1}),1\otimes
c(u_{2})]=0.$ (3.4)

###  3.2 Lax algebra and generalized Gaudin models in external field

Using the classical $r$-matrix $r(u_{1},u_{2})$ it is possible to define the
tensor Lie–Poisson bracket in the space of certain $\mathfrak{gl}(2)$-valued
functions of the complex parameter $u$ [4, 5, 13]:

$\\{{L}(u_{1})\otimes
1,1\otimes{L}(u_{2})\\}=[r_{12}(u_{1},u_{2}),{L}(u_{1})\otimes
1]-[r_{21}(u_{2},u_{1}),1\otimes{L}(u_{2})],$ (3.5)

where

$L(u)=\sum\limits_{i,j=1}^{2}L_{ij}(u)X_{ij}.$

The tensor bracket (3.5) guarantees commutativity of spectral invariants of
the Lax matrix

$\big{\\{}\operatorname{tr}L^{m}(u),\operatorname{tr}{L}^{n}(v)\big{\\}}=0$

and permits one to define a completely integrable Hamiltonian system.

For a given classical $r$-matrix there exist different types of the dependence
of $L(u)$ on spectral parameter and dynamical variables [24, 25, 26]. In the
present paper we will consider the dependence leading to Gaudin-type models in
an external magnetic field [20, 21, 22].

In more details, let ${S}_{ij}^{(m)}$, $i,j=1,\dots,n$, $m=1,\dots,N$ be
linear coordinate functions on the dual space to the Lie algebra
$\mathfrak{gl}(2)^{\oplus N}$ with the following Poisson brackets

$\big{\\{}{S}_{ij}^{(m)},{S}_{kl}^{(p)}\big{\\}}=\delta^{pm}\big{(}\delta_{kj}{S}_{il}^{(m)}-\delta_{il}{S}_{kj}^{(m)}\big{)}.$
(3.6)

Let us fix $N$ distinct points of the complex plane $\nu_{m}$, $m=1,\dots,N$.
As a consequence of the generalized classical Yang–Baxter equation (3.1) and
of the shift equation (3.4), it is possible to introduce the following
classical Lax operator [20, 21, 22]:

${L}(u)=\sum\limits_{i,j=1}^{2}{L}_{ij}(u)X_{ij}\equiv\sum\limits_{m=1}^{N}\sum\limits_{i,j,p,q=1}^{2}r^{ij,pq}(\nu_{m},u){S}^{(m)}_{ij}X_{pq}+c(u),$
(3.7)

where $c(u)=\sum\limits_{i,j=1}^{2}c_{ij}(u)X_{ij}$ is a solution of the shift
equation (3.4).

The corresponding Poisson-commuting Gaudin-type Hamiltonians read as follows

$\displaystyle{H}_{l}=\sum\limits_{k=1,k\neq
l}^{N}\sum\limits_{i,j,p,q=1}^{n}r^{ij,pq}(\nu_{k},\nu_{l}){S}_{ij}^{(k)}{S}_{pq}^{(l)}+\sum\limits_{i,j,p,q=1}^{n}r_{0}^{ij,pq}(\nu_{l},\nu_{l}){S}_{ij}^{(l)}{S}_{pq}^{(l)}+\sum\limits_{i,j=1}^{n}{c}_{ij}(\nu_{l})S^{(l)}_{ij},$

where $r_{0}^{ij,pq}(u,v)$ are the matrix elements of the regular part of the
$r$-matrix and ${c}_{ij}(\nu_{l})$ play the role of the components of external
magnetic field.

######  Remark 3.5.

The brackets (3.6) possess the following Casimir functions

$I_{1i}=\sum\limits_{i=1}^{N}\big{(}S^{(i)}_{11}+S^{(i)}_{22}\big{)},\qquad
I_{2i}=\sum\limits_{i=1}^{N}\big{(}S^{(i)}_{11}S^{(i)}_{22}-S^{(i)}_{12}S^{(i)}_{21}\big{)},\qquad
i=1,\dots,N.$

###  3.3 Quasi-trigonometric case

In this subsection we will illustrate the material of the previous subsections
by the concrete example of $\mathfrak{gl}(2)\otimes\mathfrak{gl}(2)$-valued
classical $r$-matrices and the corresponding Gaudin models.

####  3.3.1 Quasi-trigonometric $\boldsymbol{r}$-matrices

Let us consider the following $\mathfrak{gl}(2)\otimes\mathfrak{gl}(2)$-valued
function of two complex variables

$\displaystyle
r(u,v)=\left(\frac{1}{2}\frac{(u+v)}{(v-u)}+c_{1}\right)X_{11}\otimes
X_{11}+\left(\frac{1}{2}\frac{(u+v)}{(v-u)}+c_{2}\right)X_{22}\otimes X_{22}$
$\displaystyle\hphantom{r(u,v)=}{}+\frac{v}{(v-u)}X_{12}\otimes
X_{21}+\frac{u}{(v-u)}X_{21}\otimes X_{12}.$ (3.8)

It is possible to show [23] that the function (3.8) satisfies generalized
classical Yang–Baxter equation (3.1), i.e., is a classical $r$-matrix. Using
the trigonometric parametrization $u={\rm e}^{\phi}$, $v={\rm e}^{\psi}$ it is
easy to show that the $r$-matrix (3.8) satisfies the condition (3.2).

We will call the $r$-matrix (3.8) to be quasi-trigonometric.

It is easy to show that the $r$-matrix (3.8) possess a symmetry (3.3) with
respect to the Cartan-subalgebra of the diagonal matrices.

It is also easy to show that the shift element for the $r$-matrix (3.8)
satisfying the equation (3.4) has the following form

$c(u)=2(c_{11}X_{11}+c_{22}X_{22}),$ (3.9)

where the overall coefficient (two) is introduced for the further notational
convenience.

######  Remark 3.6.

In the case $c_{1}=0$, $c_{2}=0$ the $r$-matrix (3.8) coincides with skew-
symmetric trigonometric $r$-matrix. In all other cases this $r$-matrix is not
skew-symmetric.

######  Remark 3.7.

In the present paper we will concentrate on the one-parametric sub-family of
the $r$-matrices (3.8) characterized by the property $c_{2}=-c_{1}$. This
family evidently includes a skew-symmetric point $c_{1}=c_{2}=0$ as a special
partial case.

####  3.3.2 Quasi-trigonometric Lax algebra and Gaudin-type-models

The Lax algebra that correspond to the $r$-matrix (3.8) usually possesses [26]
the function $M$ associated with the Cartan-invariance of the $r$-matrix
(3.8), that satisfies the following equality

$\displaystyle\\{M,L_{ij}(u)\\}=\operatorname{sign}(j-i)(1-\delta_{ij})L_{ij}(u).$
(3.10)

It is easy to show [26] that $M$ is an integral of motion, i.e.,

$\big{\\{}M,\operatorname{tr}L^{k}(u)\big{\\}}=0,\qquad\forall\,k\in\mathbb{N}.$

As we will show below this additional integral is important for a separation
of variables.

The Lax matrix of the Gaudin-type models in an external magnetic field (3.7)
corresponding to the $r$-matrix (3.8) has the following form

$\displaystyle
L(v)=\sum\limits_{i=1}^{N}\left(\left(\frac{1}{2}\frac{\nu_{i}+v}{\nu_{i}-v}+c_{1}\right)S^{(i)}_{11}X_{11}+\left(\frac{1}{2}\frac{\nu_{i}+v}{\nu_{i}-v}+c_{2}\right)S^{(i)}_{22}X_{22}\right.$
$\displaystyle\left.\hphantom{L(v)=\sum\limits_{i=1}^{N}}{}+\frac{vS^{(i)}_{12}}{(\nu_{i}-v)}X_{21}+\frac{\nu_{i}S^{(i)}_{21}}{(\nu_{i}-v)}X_{12}\right)+c(v),$
(3.11)

where $c(v)$ is given by the formula (3.9).

The Poisson-commuting Gaudin-type Hamiltonians in an external magnetic field
are

$H_{j}=\frac{1}{2}\operatorname{res}_{v=\nu_{j}}\operatorname{tr}L^{2}(v),\qquad
j=1,\dots,N.$

They have the following explicit form

$\displaystyle H_{j}=\sum\limits_{i=1,i\neq
j}^{N}\left(\left(\frac{1}{2}\frac{\nu_{i}+\nu_{j}}{\nu_{i}-\nu_{j}}+c_{1}\right)S^{(i)}_{11}S^{(j)}_{11}+\left(\frac{1}{2}\frac{\nu_{i}+\nu_{j}}{\nu_{i}-\nu_{j}}+c_{2}\right)S^{(i)}_{22}S^{(j)}_{22}+\frac{\nu_{j}S^{(i)}_{12}S^{(j)}_{21}}{(\nu_{i}-\nu_{j})}\right.$
$\displaystyle\left.\hphantom{H_{j}=}{}+\frac{\nu_{i}S^{(i)}_{21}S^{(j)}_{12}}{(\nu_{i}-\nu_{j})}\right)+c_{1}(S^{(j)}_{11})^{2}+c_{2}(S^{(j)}_{22})^{2}+2c_{11}S^{(j)}_{11}+2c_{22}S^{(j)}_{22},\qquad
j=1,\dots,N.\\!\\!\\!$ (3.12)

There additional linear integral $M$ commutes with all the functions $H_{j}$
and has the form

$M=\frac{1}{2}\sum\limits_{j=1}^{N}\big{(}S^{(j)}_{22}-S^{(j)}_{11}\big{)}.$

######  Remark 3.8.

Observe that the Hamiltonians (3.12) differ from the standard trigonometric
Gaudin Hamiltonians by “dynamical”, i.e., dependent on the integral $M$ and
linear Casimir functions, magnetic fields [27]. By the other words, all the
corresponding “quasi-trigonometric” models are very close to the trigonometric
ones, but not equivalent to them, in particular because the corresponding
quasi-trigonometric $r$-matrices are not equivalent to the trigonometric one.

##  4 Separation of variables: quasi-trigonometric case

Let us consider the case of the Lax-integrable models governed by quasi-
trigonometric $r$-matrix (3.8). It occurred that in the case $c_{2}=-c_{1}$
for such the models there exist two one-parametric families of separating
functions $A(u)$, $B(u)$. We will consider them one by one in the next
subsections.

###  4.1 The first family of separated variables

Let us consider the following linear in the matrix elements of the Lax matrix
functions

$\displaystyle
B(u)=L_{12}(u)+\frac{k}{u}(L_{11}(u)-L_{22}(u))-\frac{k^{2}}{u^{2}}L_{21}(u),$
(4.1a) $\displaystyle
A(u)=L_{22}(u)+\frac{k}{u}L_{21}(u)+\left(c_{1}+\frac{1}{2}\right)M,$ (4.1b)
$\displaystyle C(u)=L_{21}(u).$ (4.1c)

The following proposition holds true.

######  Proposition 4.1.

Let the $\mathfrak{gl}(2)$-valued Lax matrix $L(u)$ satisfy tensor Poisson
bracket (3.5) with the $r$-matrix (3.8). Let $c_{2}=-c_{1}$. Then the
functions $B(u)$ and $A(u)$ satisfy the following Poisson algebra

$\displaystyle\\{B(u),B(v)\\}=\frac{k}{uv}(uB(u)-vB(v)),$ (4.2a)
$\displaystyle\\{A(u),B(v)\\}=\frac{1}{u-v}(uB(u)-vB(v)),$ (4.2b)
$\displaystyle\\{A(u),A(v)\\}=0.$ (4.2c)

###### Proof.

The proposition is proven by the direct calculation using the explicit form of
the functions $A(u)$, $B(v)$, the explicit form of the classical $r$-matrix,
the Poisson brackets (3.5) and the relation (3.10). ∎

Comparing the algebra (4.2) with the algebra (2.2) we obtain that the
relations (2.2a)–(2.2c) are automatically satisfied and the relation (2.3) is
satisfied after the transformation $A(u)\rightarrow\frac{A(u)}{u}$. That is
why introducing the coordinates $x_{j}$, $p_{j}$ by the relations

$B(x_{j})=0,\qquad p_{j}=\frac{A(x_{j})}{x_{j}}$

we obtain that these coordinates are canonical

$\\{x_{i},x_{j}\\}=0,\qquad\\{p_{i},p_{j}\\}=0,\qquad\\{x_{j},p_{i}\\}=\delta_{ij}.$

Hence we have constructed a set of canonical coordinates associated with the
quasi-trigonometric $r$-matrices. Now it is necessary to show that they
satisfy the equations of separation.

The following proposition holds true.

######  Proposition 4.2.

The functions $A(u)$, $B(u)$, $C(u)$ satisfy the following algebraic relation

$\det\left(L(u)-\left(A(u)-\left(c_{1}+\frac{1}{2}\right)M\right){\rm
Id}\right)+B(u)C(u)=0.$ (4.3)

###### Proof.

The proposition is proven by the direct calculation using the explicit form of
the functions $A(u)$, $B(v)$, $C(u)$. ∎

From the relation (4.3) follows that we have obtained the equations of
separation for the trigonometric models which can be written more explicitly
as follows

$\det\left(L(x_{j})-\left(x_{j}p_{j}-\left(c_{1}+\frac{1}{2}\right)M\right){\rm
Id}\right)=0.$ (4.4)

Now in order to state that the obtained coordinates are the separated
coordinates indeed, it is necessary to show that the set of the constructed
coordinates is complete on symplectic leaves of the Lie–Poisson brackets. The
following proposition holds true.

######  Proposition 4.3.

Let $L(u)$ be the Lax matrix of the $N$-spin Gaudin model (3.11). Let $k\neq
0$. Then the function $B(u)$ given by the formula (4.1a) possesses $N$ non-
constant zeros.

###### Proof.

The proposition is proven by a direct calculation. Using the explicit form
(4.1a) of the function $B(u)$ and the Lax matrix (3.11) it is easy to show
that it is a rational function in $u$, nominator of which is a polynomial of
degree $N$ whose constant in $u$ term is not zero if $k\neq 0$.

Now, using the fact that symplectic leafs of the Lie–Poisson brackets in
$\mathfrak{gl}(2)^{\oplus N}$ are $2N$-dimensional we obtain that our
construction produce the needed number of the canonical coordinates satisfying
the equations of separation (4.4), i.e., produce a complete set of the
coordinates of separation. ∎

######  Remark 4.4.

Observe that the constructed set of the separated coordinates is complete even
without external magnetic field, i.e., when $c_{11}=c_{22}=0$.

######  Remark 4.5.

Observe, that $B(u)=L_{12}^{K}(u)$, where $L^{K}(u)=K(u)^{-1}L(u)K(u)$ and

$K(u)=\left(\begin{matrix}1&\dfrac{k}{u}\vspace{1mm}\\\
0&1\end{matrix}\right).$

The element $A(u)$ coincides with $L_{22}^{K}(u)$ only in the case
$c_{1}=-\frac{1}{2}$. Consequently the equations of separation (4.4) do not
coincide with the spectral curve of the Lax matrix if $c_{1}\neq-\frac{1}{2}$.

###  4.2 The second family of separated variables

Let us consider the following linear in the matrix elements of the Lax matrix
functions

$\displaystyle B(u)=L_{12}(u)+{k}(L_{11}(u)-L_{22}(u))-{k^{2}}L_{21}(u),$
(4.5a) $\displaystyle
A(u)=L_{11}(u)-{k}L_{21}(u)+\left(c_{1}+\frac{1}{2}\right)M,$ (4.5b)
$\displaystyle C(u)=L_{21}(u).$ (4.5c)

The following proposition holds true.

######  Proposition 4.6.

Let the $\mathfrak{gl}(2)$-valued Lax matrix $L(u)$ satisfy tensor Poisson
bracket (3.5) with the $r$-matrix (3.8). Let $c_{2}=-c_{1}$. Then the
functions $B(u)$ and $A(u)$ satisfy the following Poisson algebra:

$\displaystyle\\{B(u),B(v)\\}=-{k}(B(u)-B(v)),$ (4.6a)
$\displaystyle\\{A(u),B(v)\\}=-\frac{u}{u-v}(B(u)-B(v)),$ (4.6b)
$\displaystyle\\{A(u),A(v)\\}=0.$ (4.6c)

###### Proof.

The proposition is proven by the direct calculation using the explicit form of
the functions $A(u)$, $B(v)$, the explicit form of the classical $r$-matrix,
the Poisson brackets (3.5) and the relation (3.10). ∎

Comparing the algebra (4.6) with the algebra (2.2) we obtain that the
relations (2.2a)–(2.2c) are automatically satisfied and the relation (2.3) is
satisfied after the transformation $A(u)\rightarrow-\frac{A(u)}{u}$. That is
why introducing the coordinates $x_{j}$, $p_{j}$ by the relations

$B(x_{j})=0,\qquad p_{j}=-\frac{A(x_{j})}{x_{j}}$

we obtain that these coordinates are canonical

$\\{x_{i},x_{j}\\}=0,\qquad\\{p_{i},p_{j}\\}=0,\qquad\\{x_{j},p_{i}\\}=\delta_{ij}.$

Hence we have constructed a set of canonical coordinates associated with the
quasi-trigonometric $r$-matrices. Now it is necessary to show that they
satisfy some equations of separation.

The following proposition holds true.

######  Proposition 4.7.

The functions $A(u)$, $B(u)$, $C(u)$ satisfy the following algebraic relation

$\det\left(L(u)-\left(A(u)-\left(c_{1}+\frac{1}{2}\right)M\right){\rm
Id}\right)+B(u)C(u)=0.$ (4.7)

###### Proof.

The proposition is proven by the direct calculation using the explicit form of
the functions $A(u)$, $B(v)$, $C(u)$. ∎

From the relation (4.7) it follows that we have obtained new equations of
separation for the trigonometric models which can be written explicitly as
follows

$\det\left(L(x_{j})+\left(x_{j}p_{j}+\left(c_{1}+\frac{1}{2}\right)M\right){\rm
Id}\right)=0.$ (4.8)

Now, in order to state that the obtained coordinates are indeed the separated
coordinates it is necessary only to show that the set of the constructed
coordinates is complete on symplectic leafs of the Lie–Poisson brackets. The
following proposition holds true.

######  Proposition 4.8.

Let $L(u)$ be the Lax matrix of the $N$-spin Gaudin model (3.11). Let $k\neq
0$. Then the function $B(u)$ given by the formula (4.1a) possesses $N$ non-
constant zeros.

###### Proof.

The proposition is proven by a direct calculation. Using the explicit form of
the function $B(u)$ and the Lax matrix (3.11) it is easy to show that it is a
rational function in $u$, nominator of which is a polynomial of degree $N$
whose higher order term is non-zero if $k\neq 0$.

Now, using the fact that symplectic leafs of the Lie–Poisson brackets in
$\mathfrak{gl}(2)^{\oplus N}$ are $2N$-dimensional we obtain that our
construction produces the needed number of the canonical coordinates
satisfying the equations of separation (4.8), i.e., produces a complete set of
the coordinates of separation. ∎

######  Remark 4.9.

Observe that, similar to the first set of separated variables, the constructed
second set of separated variables is complete even without external magnetic
field.

######  Remark 4.10.

Observe, that $B(u)=L_{12}^{K}(u)$, where $L^{K}(u)=K^{-1}L(u)K$ and

$K=\left(\begin{matrix}1&k\\\ 0&1\end{matrix}\right).$

The element $A(u)$ coincide with $L_{11}^{K}(u)$ only in the case
$c_{1}=-\frac{1}{2}$. Consequently the equations of separation (4.8) do not
coincide with the spectral curve of the Lax matrix unless
$c_{1}=-\frac{1}{2}$.

##  5 Example: $\boldsymbol{N=2}$ quasi-trigonometric Gaudin model

In this section we will illustrate the results of the previous section by the
example of $N=2$ quasi-trigonometric Gaudin models.

###  5.1 $\boldsymbol{N=2}$ quasi-trigonometric Gaudin model

Let us consider the Hamiltonians and Lax operators of $N=2$ Gaudin-type models
more explicitly. Let $S^{(m)}_{ij}$, $i,j,m=1,2$ be coordinate functions on
$(\mathfrak{gl}(2)\oplus\mathfrak{gl}(2))^{*}$ with the standard Lie–Poisson
brackets (3.6). Here-after for the further notational convenience we will use
the following new notations

$S^{(1)}_{ij}\equiv S_{ij},\qquad S^{(2)}_{ij}\equiv T_{ij}.$

The above Poisson brackets have two linear Casimir functions

$I_{11}=S_{11}+S_{22},\qquad I_{12}=T_{11}+T_{22}$

and two quadratic ones

$I_{21}=S_{11}S_{22}-S_{12}S_{21},\qquad I_{22}=T_{11}T_{22}-T_{12}T_{21}.$

The Lax operator $L(v)$ of two-spin quasi-trigonometric Gaudin model in an
external magnetic field is written as follows

$\displaystyle L(v)=L^{S}(v)+L^{T}(v)+c(v),$ (5.1a) $\displaystyle
L^{S}(v)=\left(\frac{1}{2}\frac{\nu_{1}+v}{\nu_{1}-v}+c_{1}\right)S_{11}X_{11}+\left(\frac{1}{2}\frac{\nu_{1}+v}{\nu_{1}-v}+c_{2}\right)S_{22}X_{22}$
$\displaystyle\hphantom{L^{S}(v)=}{}+\frac{vS_{12}}{(\nu_{1}-v)}X_{21}+\frac{\nu_{1}S_{21}}{(\nu_{1}-v)}X_{12},$
(5.1b) $\displaystyle
L^{T}(v)=\left(\frac{1}{2}\frac{\nu_{2}+v}{\nu_{2}-v}+c_{1}\right)T_{11}X_{11}+\left(\frac{1}{2}\frac{\nu_{2}+v}{\nu_{2}-v}+c_{2}\right)T_{22}X_{22}$
$\displaystyle\hphantom{L^{T}(v)=}{}+\frac{vT_{12}}{(\nu_{2}-v)}X_{21}+\frac{\nu_{2}T_{21}}{(\nu_{2}-v)}X_{12},$
(5.1c) $\displaystyle c(v)=2c_{11}X_{11}+2c_{22}X_{22}.$ (5.1d)

The corresponding Gaudin-type Hamiltonians in a magnetic field have the
following form

$\displaystyle
H_{1}=\left(\frac{1}{2}\frac{(\nu_{1}+\nu_{2})}{(\nu_{2}-\nu_{1})}+c_{1}\right)T_{11}S_{11}+\left(\frac{1}{2}\frac{(\nu_{1}+\nu_{2})}{(\nu_{2}-\nu_{1})}+c_{2}\right)T_{22}S_{22}$
$\displaystyle\hphantom{H_{1}=}{}+\frac{\nu_{1}T_{12}S_{21}}{(\nu_{2}-\nu_{1})}+\frac{\nu_{2}T_{21}S_{12}}{(\nu_{2}-\nu_{1})}+c_{1}S^{2}_{11}+c_{2}S^{2}_{22}+2c_{11}S_{11}+2c_{22}S_{22},$
$\displaystyle
H_{2}=\left(\frac{1}{2}\frac{(\nu_{1}+\nu_{2})}{(\nu_{1}-\nu_{2})}+c_{1}\right)S_{11}T_{11}+\left(\frac{1}{2}\frac{(\nu_{1}+\nu_{2})}{(\nu_{1}-\nu_{2})}+c_{2}\right)S_{22}T_{22}$
$\displaystyle\hphantom{H_{2}=}{}+\frac{\nu_{2}S_{12}T_{21}}{(\nu_{1}-\nu_{2})}+\frac{\nu_{1}S_{21}T_{12}}{(\nu_{1}-\nu_{2})}+c_{1}T^{2}_{11}+c_{2}T^{2}_{22}+2c_{11}T_{11}+2c_{22}T_{22}.$

The additional linear integral $M$ associated with the Cartan-symmetry of the
$r$-matrix is

$M=-\frac{1}{2}(S_{11}-S_{22}+T_{11}-T_{22}).$

It is not independent – it is related with $H_{1}$, $H_{2}$ and linear Casimir
functions as follows

$\displaystyle
H_{1}+H_{2}+2c_{1}(I_{11}+I_{12})M+c_{11}(2M-I_{11}-I_{12})-c_{22}(2M+I_{11}+I_{12})$
$\displaystyle\qquad{}-\left(\frac{1}{4}(c_{1}+c_{2})\right)(2M+I_{11}+I_{12})^{2}=0.$

In the interesting for us case $c_{2}=-c_{1}$ we obtain that

$M=\frac{1}{2}\frac{((c_{11}+c_{22})I_{11}+(c_{11}+c_{22})I_{12}-H_{1}-H_{2})}{((I_{11}+I_{12})c_{1}+c_{11}-c_{22})}.$

######  Remark 5.1.

Hereafter we will consider the case of the $r$-matrices with $c_{2}=-c_{1}$
only. For the purpose of convenience will chose the following poles of the Lax
matrix

$\nu_{2}=-\nu_{1}.$

In order to simplify all the formulae we will restrict ourselves to the
consideration of the traceless case, i.e., we will hereafter put $I_{11}=0$,
$I_{12}=0$.

###  5.2 The first family of separated variables

####  5.2.1 The separating functions

The first family of the separated variables are given by the separating
functions (4.1a)–(4.1b). Let us specify them for the Lax matrix (5.1). We will
have

$\displaystyle
B(u)=\frac{1}{2u(u-\nu_{1})(u+\nu_{1})}\bigl{(}\bigl{(}k(2c_{1}-1)S_{11}-2\nu_{1}S_{21}+k(2c_{1}+1)S_{22}+k(2c_{1}-1)T_{11}$
$\displaystyle\hphantom{B(u)=}{}+2T_{21}\nu_{1}+k(2c_{1}+1)T_{22}+4k(-c_{22}+c_{11})\bigr{)}u^{2}+\bigl{(}2k^{2}T_{12}+2S_{12}k^{2}$
$\displaystyle\hphantom{B(u)=}{}+2k\nu_{1}S_{22}+2k\nu_{1}T_{11}-2k\nu_{1}T_{22}-2\nu_{1}^{2}S_{21}-2\nu_{1}^{2}T_{21}-2k\nu_{1}S_{11}\bigr{)}u$
$\displaystyle\hphantom{B(u)=}{}-k\bigl{(}\nu_{1}^{2}(2c_{1}+1)S_{11}+2\nu_{1}S_{12}k-\nu_{1}^{2}(2c_{1}-1)S_{22}-\nu_{1}^{2}(2c_{1}+1)T_{11}-2\nu_{1}T_{12}k$
$\displaystyle\hphantom{B(u)=}{}-\nu_{1}^{2}(2c_{1}-1)T_{22}-4\nu_{1}^{2}(-c_{22}+c_{11})\bigr{)}\bigr{)},$
$\displaystyle
A(u)=\frac{1}{4(u^{2}-\nu_{1}^{2})}\bigl{(}\bigl{(}{-}(1+2c_{1})S_{11}-(1+2c_{1})S_{22}-(1+2c_{1})T_{11}-(1+2c_{1})T_{22}$
$\displaystyle\hphantom{A(u)=}{}+8c_{22}\bigr{)}u^{2}+\bigl{(}{-}4S_{12}k+4T_{22}\nu_{1}-4T_{12}k-4S_{22}\nu_{1}\bigr{)}u+\nu_{1}^{2}(2c_{1}+1)S_{11}-4kS_{12}\nu_{1}$
$\displaystyle\hphantom{A(u)=}{}+\nu_{1}^{2}(2c_{1}-3)S_{22}+\nu_{1}^{2}(2c_{1}+1)T_{11}+4kT_{12}\nu_{1}+\nu_{1}^{2}(2c_{1}-3)T_{22}-8c_{22}\nu_{1}^{2}\bigr{)}.$

The coordinates of separation are two solutions $x_{1}$, $x_{2}$ of the
quadratic in $u$ equation

$B(x_{i})=0.$

The canonically conjugated momenta are given by the formula

$p_{i}=\frac{A(x_{i})}{x_{i}},\qquad i=1,2.$

######  Remark 5.2.

Observe that in the case $k=0$ the polynomial $B(u)$ has only one non-constant
root and, that is why, can not be used in our construction of separated
variables.

####  5.2.2 The equations of separation and Abel-type equations

In terms of the Hamiltonians $H_{1}$, $H_{2}$, the Casimir functions $I_{21}$,
$I_{22}$ and the canonical coordinates $x_{i}$, $p_{i}$ the curves of
separation (4.4) have the following form

$\displaystyle\Phi(x_{i},p_{i},H_{1},H_{2},I_{21},I_{22})=x_{i}^{2}p_{i}^{2}+\left(\frac{H_{1}+H_{2}}{2(c_{11}-c_{22})}-2(c_{11}+c_{22})\right)x_{i}p_{i}+\frac{x_{i}\nu_{1}I_{21}}{(\nu_{1}-x_{i})^{2}}$
$\displaystyle-\frac{x_{i}\nu_{1}I_{22}}{(x_{i}+\nu_{1})^{2}}-\frac{(\nu_{1}c_{11}-x_{i}c_{22})H_{1}}{(c_{11}-c_{22})(\nu_{1}-x_{i})}-\frac{(\nu_{1}c_{11}+x_{i}c_{22})H_{2}}{(c_{11}-c_{22})(x_{i}+\nu_{1})}+4c_{11}c_{22}=0,\qquad
i=1,2.\\!\\!\\!$ (5.2)

Using either equations of separation (5.2) or the reconstruction formulas it
is possible to express $H_{1}$, $H_{2}$ via the coordinates of separation
$x_{i}$, $p_{i}$, $i=1,2$ and the Casimir functions.

Taking into account the canonical Poisson brackets

$\\{x_{i},p_{j}\\}=\delta_{ij},\qquad\\{x_{i},x_{j}\\}=0,\qquad\\{p_{i},p_{j}\\}=0,\qquad
i,j=1,2,$

calculating with their help the time derivatives of the coordinates of
separation

$\frac{{\rm d}x_{i}}{{\rm d}t_{j}}\equiv\\{x_{i},H_{j}\\}$

and making simple transformations we obtain Abel-type equations in the
differential form

$\displaystyle\frac{(x_{1}p_{1}(x_{1}-\nu_{1})+2c_{11}\nu_{1}-2c_{22}x_{1}){\rm
d}x_{1}}{\big{(}4x_{1}p_{1}(c_{11}-c_{22})+H_{1}+H_{2}-4\big{(}c_{11}^{2}-c_{22}^{2}\big{)}\big{)}x_{1}(x_{1}-\nu_{1})}$
$\displaystyle\qquad{}+\frac{(x_{2}p_{2}(x_{2}-\nu_{1})+2c_{11}\nu_{1}-2c_{22}x_{2}){\rm
d}x_{2}}{\big{(}4x_{2}p_{2}(c_{11}-c_{22})+H_{1}+H_{2}-4\big{(}c_{11}^{2}-c_{22}^{2}\big{)}\big{)}x_{2}(x_{2}-\nu_{1})}=-{\rm
d}t_{1},$
$\displaystyle\frac{(x_{1}p_{1}(x_{1}+\nu_{1})-2c_{11}\nu_{1}-2c_{22}x_{1}){\rm
d}x_{1}}{\big{(}4x_{1}p_{1}(c_{11}-c_{22})+H_{1}+H_{22}-4\big{(}c_{11}^{2}-c_{22}^{2}\big{)}\big{)}x_{1}(x_{1}+\nu_{1})}$
$\displaystyle\qquad{}+\frac{(x_{2}p_{2}(x_{2}+\nu_{1})-2c_{11}\nu_{1}-2c_{22}x_{2}){\rm
d}x_{2}}{\big{(}4x_{2}p_{2}(c_{11}-c_{22})+H_{1}+H_{22}-4\big{(}c_{11}^{2}-c_{22}^{2}\big{)}\big{)}x_{2}(x_{2}+\nu_{1})}=-{\rm
d}t_{2},$

where the momenta $p_{i}$, $i=1,2$ are calculated using the curves of
separation (5.2).

###  5.3 The second family of separated variables

####  5.3.1 The separating functions

The second family of the separated variables is obtained using the separating
functions (4.5a)–(4.5b). Let us specify it for the Lax matrix (5.1). We will
have

$\displaystyle
B(u)=\frac{1}{2\big{(}u^{2}-\nu_{1}^{2}\big{)}}\bigl{(}k\bigl{(}(2c_{1}-1)S_{11}+2kS_{12}+k(2c_{1}+1)S_{22}+(2c_{1}-1)T_{11}+2kT_{12}$
$\displaystyle\hphantom{B(u)=}{}+(2c_{1}+1)T_{22}+4(c_{11}-c_{22})\bigr{)}u^{2}+\nu_{1}\bigl{(}2k^{2}S_{12}+2kT_{11}+2kS_{22}+2T_{21}-2k^{2}T_{12}$
$\displaystyle\hphantom{B(u)=}{}-S_{21}-2kS_{11}-2kT_{22}\bigr{)}u-\nu_{1}^{2}\big{(}k(2c_{1}+1)S_{11}-2S_{21}-k(2c_{1}-1)S_{22}$
$\displaystyle\hphantom{B(u)=}{}-k(2c_{1}+1)T_{11}-2T_{21}-k(2c_{1}-1)T_{22}-4k(c_{11}-c_{22})\big{)}\bigr{)},$
$\displaystyle
A(u)=\frac{1}{4\big{(}\nu_{1}^{2}-u^{2}\big{)}}\bigl{(}\bigl{(}(3-2c_{1})S_{11}-4S_{12}k+(-2c_{1}-1)S_{22}+(3-2c_{1})T_{11}-4T_{12}k$
$\displaystyle\hphantom{A(u)=}{}+(-2c_{1}-1)T_{22}-8c_{11}\bigr{)}u^{2}+(4S_{11}\nu_{1}-4T_{11}\nu_{1}-4kS_{12}\nu_{1}+4kT_{12}\nu_{1})u$
$\displaystyle\hphantom{A(u)=}{}+\nu_{1}^{2}(2c_{1}+1)S_{11}+\nu_{1}^{2}(2c_{1}+1)S_{22}+\nu_{1}^{2}(2c_{1}+1)T_{11}+\nu_{1}^{2}(2c_{1}+1)T_{22}+8c_{11}\nu_{1}^{2}\bigr{)}.$

The coordinates of separation are two solutions $x_{1}$, $x_{2}$ of the
quadratic in $u$ equation

$B(x_{i})=0.$

The canonically conjugated momenta are given by the formula

$p_{i}=-\frac{A(x_{i})}{x_{i}},\qquad i=1,2.$

######  Remark 5.3.

Observe that in the case $k=0$ the polynomial $B(u)$ has only one root and,
that is why, can not be used in our construction of separated variables.

####  5.3.2 The Abel-type equations

In terms of the Hamiltonians $H_{1}$, $H_{2}$ and Casimir functions $I_{21}$,
$I_{22}$ and the canonical coordinates $x_{i}$, $p_{i}$ the curves of the
separation (4.8) have the following form

$\displaystyle\Phi(x_{i},p_{i},H_{1},H_{2},I_{21},I_{22})=x_{i}^{2}p_{i}^{2}-\left(\frac{H_{1}+H_{2}}{2(c_{11}-c_{22})}-2(c_{11}+c_{22})\right)x_{i}p_{i}+\frac{x_{i}\nu_{1}I_{21}}{(\nu_{1}-x_{i})^{2}}$
$\displaystyle-\frac{x_{i}\nu_{1}I_{22}}{(x_{i}+\nu_{1})^{2}}-\frac{(\nu_{1}c_{11}-x_{i}c_{22})H_{1}}{(c_{11}-c_{22})(\nu_{1}-x_{i})}-\frac{(\nu_{1}c_{11}+x_{i}c_{22})H_{2}}{(c_{11}-c_{22})(x_{i}+\nu_{1})}+4c_{11}c_{22}=0,\qquad
i=1,2.\\!\\!\\!$ (5.3)

Using either equations of separation (5.3) or the reconstruction formulae it
is possible to express $H_{1}$, $H_{2}$ via the coordinates of separation
$x_{i}$, $p_{i}$, $i=1,2$ and the Casimir functions.

Taking into account canonical commutation relations

$\\{x_{i},p_{j}\\}=\delta_{ij},\qquad\\{x_{i},x_{j}\\}=0,\qquad\\{p_{i},p_{j}\\}=0,\qquad
i,j=1,2,$

calculating with their help the time derivatives of the coordinates of
separation

$\frac{{\rm d}x_{i}}{{\rm d}t_{j}}\equiv\\{x_{i},H_{j}\\},\qquad i,j=1,2$

and making simple transformations we obtain the equations of motion in the
Abel-type form

$\displaystyle\frac{(-x_{1}p_{1}(x_{1}-\nu_{1})+2c_{11}\nu_{1}-2c_{22}x_{1}){\rm
d}x_{1}}{\big{(}4x_{1}p_{1}(c_{11}-c_{22})-H_{1}-H_{2}+4\big{(}c_{11}^{2}-c_{22}^{2}\big{)}\big{)}x_{1}(x_{1}-\nu_{1})}$
$\displaystyle\qquad{}+\frac{(-x_{2}p_{2}(x_{2}-\nu_{1})+2c_{11}\nu_{1}-2c_{22}x_{2}){\rm
d}x_{2}}{\big{(}4x_{2}p_{2}(c_{11}-c_{22})-H_{1}-H_{2}+4\big{(}c_{11}^{2}-c_{22}^{2}\big{)}\big{)}x_{2}(x_{2}-\nu_{1})}=-{{\rm
d}t_{1}},$
$\displaystyle\frac{(-x_{1}p_{1}(x_{1}+\nu_{1})-2c_{11}\nu_{1}-2c_{22}x_{1}){\rm
d}x_{1}}{\big{(}4x_{1}p_{1}(c_{11}-c_{22})-H_{1}-H_{22}+4\big{(}c_{11}^{2}-c_{22}^{2}\big{)}\big{)}x_{1}(x_{1}+\nu_{1})}$
$\displaystyle\qquad{}+\frac{(-x_{2}p_{2}(x_{2}+\nu_{1})-2c_{11}\nu_{1}-2c_{22}x_{2}){\rm
d}x_{2}}{\big{(}4x_{2}p_{2}(c_{11}-c_{22})-H_{1}-H_{22}+4\big{(}c_{11}^{2}-c_{22}^{2}\big{)}\big{)}x_{2}(x_{2}+\nu_{1})}=-{{\rm
d}t_{2}},$

where the momenta $p_{i}$, $i=1,2$ are calculated using the curves of
separation (5.3).

######  Remark 5.4.

Observe, that the equations of separation and Abel-type equations for our two
families of separated variables do not coincide. The corresponding
reconstruction formulae do not coincide either (see Appendices A and B).

##  6 Conclusion and discussion

In this paper for the classical Lax-integrable Hamiltonian systems governed by
the one-parametric family of non-skew-symmetric, non-dynamical
$\mathfrak{gl}(2)\otimes\mathfrak{gl}(2)$-valued quasi-trigonometric classical
$r$-matrices we have constructed two new one-parametric families of separated
variables. We have shown that for all but one $r$-matrices in the considered
one-parametric families the corresponding curves of separation are “shifted”
spectral curves of the initial Lax matrix. We have illustrated the proposed
scheme by an example of SoV for $N=2$ quasi-trigonometric Gaudin models.

It would be interesting to specify other classes of non-skew-symmetric
classical $r$-matrices for which separation curves are the shifted spectral
curves of the Lax matrices and to perform SoV for them. The work over this
problem is now in progress.

##  Appendix A  The reconstruction formulae for the first SoV

Let us now reconstruct the dynamical variables $S_{ij}$, $T_{ij}$, $i,j=1,2$
using the variables of separation $p_{i}$, $x_{j}$ constructed in Section 5.2
and the Casimir functions. For this purpose we solve a system of eight linear-
quadratic equations on eight variables $S_{ij}$, $T_{ij}$:

$\displaystyle(x_{1}+x_{2})=-\bigl{(}2k^{2}T_{12}+2S_{12}k^{2}+2k\nu_{1}S_{22}+2k\nu_{1}T_{11}-2k\nu_{1}T_{22}-2\nu_{1}^{2}S_{21}-2\nu_{1}^{2}T_{21}$
$\displaystyle\hphantom{(x_{1}+x_{2})=}{}-2k\nu_{1}S_{11}\bigr{)}\bigl{(}k(2c_{1}-1)S_{11}-2\nu_{1}S_{21}+k(2c_{1}+1)S_{22}+k(2c_{1}-1)T_{11}$
$\displaystyle\hphantom{(x_{1}+x_{2})=}{}+2T_{21}\nu_{1}+k(2c_{1}+1)T_{22}+4k(-c_{22}+c_{11})\bigr{)}^{-1},$
(A.1a) $\displaystyle
x_{1}x_{2}=k\bigl{(}\nu_{1}^{2}(2c_{1}+1)S_{11}+2\nu_{1}S_{12}k-\nu_{1}^{2}(2c_{1}-1)S_{22}-\nu_{1}^{2}(2c_{1}+1)T_{11}-2\nu_{1}T_{12}k$
$\displaystyle\hphantom{x_{1}x_{2}=}{}-\nu_{1}^{2}(2c_{1}-1)T_{22}-4\nu_{1}^{2}(-c_{22}+c_{11})\bigr{)}\bigl{(}k(2c_{1}-1)S_{11}-2\nu_{1}S_{21}+k(2c_{1}+1)S_{22}$
$\displaystyle\hphantom{x_{1}x_{2}=}{}+k(2c_{1}-1)T_{11}+2T_{21}\nu_{1}+k(2c_{1}+1)T_{22}+4k(-c_{22}+c_{11})\bigr{)}^{-1},$
(A.1b) $\displaystyle
4\big{(}x_{1}^{2}-\nu_{1}^{2}\big{)}x_{1}p_{1}=\bigl{(}\bigl{(}-(1+2c_{1})S_{11}-(1+2c_{1})S_{22}-(1+2c_{1})T_{11}-(1+2c_{1})T_{22}$
$\displaystyle\hphantom{4\big{(}x_{1}^{2}-\nu_{1}^{2}\big{)}x_{1}p_{1}=}{}+8c_{22}\bigr{)}x_{1}^{2}+\bigl{(}-4S_{12}k+4T_{22}\nu_{1}-4T_{12}k-4S_{22}\nu_{1}\bigr{)}x_{1}+\nu_{1}^{2}(2c_{1}+1)S_{11}$
$\displaystyle\hphantom{4\big{(}x_{1}^{2}-\nu_{1}^{2}\big{)}x_{1}p_{1}=}{}-4kS_{12}\nu_{1}+\nu_{1}^{2}(2c_{1}-3)S_{22}+\nu_{1}^{2}(2c_{1}+1)T_{11}+4kT_{12}\nu_{1}$
$\displaystyle\hphantom{4\big{(}x_{1}^{2}-\nu_{1}^{2}\big{)}x_{1}p_{1}=}{}+\nu_{1}^{2}(2c_{1}-3)T_{22}-8c_{22}\nu_{1}^{2}\bigr{)},$
(A.1c) $\displaystyle
4\big{(}x_{2}^{2}-\nu_{1}^{2}\big{)}x_{2}p_{2}=\bigl{(}\bigl{(}-(1+2c_{1})S_{11}-(1+2c_{1})S_{22}-(1+2c_{1})T_{11}-(1+2c_{1})T_{22}$
$\displaystyle\hphantom{4\big{(}x_{2}^{2}-\nu_{1}^{2}\big{)}x_{2}p_{2}=}{}+8c_{22}\bigr{)}x_{2}^{2}+\bigl{(}-4S_{12}k+4T_{22}\nu_{1}-4T_{12}k-4S_{22}\nu_{1}\bigr{)}x_{2}+\nu_{1}^{2}(2c_{1}+1)S_{11}$
$\displaystyle\hphantom{4\big{(}x_{2}^{2}-\nu_{1}^{2}\big{)}x_{2}p_{2}=}{}-4kS_{12}\nu_{1}+\nu_{1}^{2}(2c_{1}-3)S_{22}+\nu_{1}^{2}(2c_{1}+1)T_{11}+4kT_{12}\nu_{1}$
$\displaystyle\hphantom{4\big{(}x_{2}^{2}-\nu_{1}^{2}\big{)}x_{2}p_{2}=}{}+\nu_{1}^{2}(2c_{1}-3)T_{22}-8c_{22}\nu_{1}^{2}\bigr{)},$
(A.1d) $\displaystyle I_{21}=S_{11}S_{22}-S_{12}S_{21},$ (A.1e) $\displaystyle
I_{22}=T_{11}T_{22}-T_{12}T_{21},$ (A.1f) $\displaystyle
I_{11}=S_{11}+S_{22},$ (A.1g) $\displaystyle I_{12}=T_{11}+T_{22}.$ (A.1h)

The following proposition is proven by the direct calculations.

######  Proposition A.1.

The system of equations (A.1) is solved with respect of the variables
$S_{ij}$, $T_{ij}$, $i,j=1,2$ as follows

$\displaystyle
S_{11}=-\frac{1}{2\nu_{1}}\bigl{(}x_{1}^{2}x_{2}(\nu_{1}-x_{2})^{2}(\nu_{1}^{2}-x_{1}^{2})^{2}p_{1}^{2}-x_{1}x_{2}(x_{1}+x_{2})(\nu_{1}+x_{2})(\nu_{1}-x_{2})^{2}(\nu_{1}+x_{1})$
$\displaystyle\hphantom{S_{11}=}{}\times(\nu_{1}-x_{1})^{2}p_{2}p_{1}+x_{1}x_{2}^{2}(\nu_{1}^{2}-x_{2}^{2})^{2}(\nu_{1}-x_{1})^{2}p_{2}^{2}+2x_{1}(x_{1}-x_{2})(\nu_{1}-x_{2})^{2}(\nu_{1}+x_{1})$
$\displaystyle\hphantom{S_{11}=}{}\times(\nu_{1}-x_{1})^{2}(c_{11}\nu_{1}-x_{2}c_{22})p_{1}-2x_{2}(x_{1}-x_{2})(\nu_{1}+x_{2})(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}$
$\displaystyle\hphantom{S_{11}=}{}\times(c_{11}\nu_{1}-x_{1}c_{22})p_{2}+4x_{1}x_{2}\nu_{1}^{3}(x_{1}-x_{2})^{2}I_{21}-4c_{11}\nu_{1}c_{22}(x_{1}-x_{2})^{2}(x_{2}-\nu_{1})^{2}$
$\displaystyle\hphantom{S_{11}=}{}\times(x_{1}-\nu_{1})^{2}\bigr{)}\bigl{(}(x_{1}-x_{2})\bigl{(}(x_{1}x_{2}(\nu_{1}+x_{1})(\nu_{1}-x_{1})^{2}(\nu_{1}-x_{2})p_{1}-x_{1}x_{2}(\nu_{1}+x_{2})$
$\displaystyle\hphantom{S_{11}=}{}\times(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})p_{2}+2(x_{1}-x_{2})(\nu_{1}-x_{2})(\nu_{1}-x_{1})(c_{11}\nu_{1}^{2}+x_{1}c_{22}x_{2})\bigr{)}\bigr{)}^{-1},$
$\displaystyle
S_{12}=-\frac{x_{1}x_{2}}{2k\nu_{1}}\bigl{(}x_{1}^{2}(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}(\nu_{1}+x_{1})^{2}p_{1}^{2}-2x_{1}x_{2}(\nu_{1}+x_{2})(\nu_{1}-x_{2})^{2}(\nu_{1}+x_{1})$
$\displaystyle\hphantom{S_{12}=}{}\times(\nu_{1}-x_{1})^{2}p_{2}p_{1}+x_{2}^{2}(\nu_{1}-x_{2})^{2}(\nu_{1}+x_{2})^{2}(\nu_{1}-x_{1})^{2}p_{2}^{2}-4x_{1}c_{22}(x_{1}-x_{2})(\nu_{1}-x_{2})^{2}$
$\displaystyle\hphantom{S_{12}=}{}\times(\nu_{1}+x_{1})(\nu_{1}-x_{1})^{2}p_{1}+4x_{2}c_{22}(x_{1}-x_{2})(x_{2}+\nu_{1})(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}p_{2}$
$\displaystyle\hphantom{S_{12}=}{}+4\nu_{1}^{4}(x_{1}-x_{2})^{2}I_{21}+4c_{22}^{2}(x_{1}-x_{2})^{2}(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}\bigr{)}\bigl{(}(\nu_{1}-x_{2})(\nu_{1}-x_{1})$
$\displaystyle\hphantom{S_{12}=}{}\times(x_{1}-x_{2})\bigl{(}x_{1}x_{2}\big{(}\nu_{1}^{2}-x_{1}^{2}\big{)}p_{1}-x_{1}x_{2}\big{(}a^{2}_{1}-x^{2}_{2}\big{)}p_{2}+2(x_{1}-x_{2})$
$\displaystyle\hphantom{S_{12}=}{}\times\big{(}c_{11}\nu_{1}^{2}+x_{1}c_{22}x_{2}\big{)}\bigr{)}\bigr{)}^{-1},$
$\displaystyle
S_{21}=\frac{k}{2\nu_{1}x_{1}x_{2}}\bigl{(}x_{1}^{2}x_{2}^{2}(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}(\nu_{1}+x_{1})^{2}p_{1}^{2}-2x_{1}^{2}x_{2}^{2}(\nu_{1}+x_{2})(\nu_{1}-x_{2})^{2}$
$\displaystyle\hphantom{S_{21}=}{}\times(\nu_{1}+x_{1})(\nu_{1}-x_{1})^{2}p_{2}p_{1}+x_{1}^{2}x_{2}^{2}(\nu_{1}-x_{2})^{2}(\nu_{1}+x_{2})^{2}(\nu_{1}-x_{1})^{2}p_{2}^{2}+4c_{11}\nu_{1}x_{1}x_{2}$
$\displaystyle\hphantom{S_{21}=}{}\times(x_{1}-x_{2})(\nu_{1}-x_{2})^{2}(\nu_{1}+x_{1})(\nu_{1}-x_{1})^{2}p_{1}-4c_{11}\nu_{1}x_{1}x_{2}(x_{1}-x_{2})(\nu_{1}+x_{2})$
$\displaystyle\hphantom{S_{21}=}{}\times(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}p_{2}+4x_{1}^{2}\nu_{1}^{2}x_{2}^{2}(x_{1}-x_{2})^{2}I_{21}+4\nu_{1}^{2}c_{11}^{2}(x_{1}-x_{2})^{2}(\nu_{1}-x_{2})^{2}$
$\displaystyle\hphantom{S_{21}=}{}\times(\nu_{1}-x_{1})^{2}\bigr{)}\bigl{(}x_{1}x_{2}(x_{1}-x_{2})(\nu_{1}-x_{2})(\nu_{1}+x_{1})(\nu_{1}-x_{1})^{2}p_{1}-x_{1}x_{2}(x_{1}-x_{2})$
$\displaystyle\hphantom{S_{21}=}{}\times(\nu_{1}+x_{2})(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})p_{2}+2(x_{1}-x_{2})^{2}(\nu_{1}-x_{2})(\nu_{1}-x_{1})$
$\displaystyle\hphantom{S_{21}=}{}\times(c_{11}\nu_{1}^{2}+x_{1}c_{22}x_{2})\bigr{)}^{-1},$
$\displaystyle S_{22}=-S_{11},$ $\displaystyle
T_{11}=-\frac{1}{2\nu_{1}}\bigl{(}-x_{1}^{2}x_{2}(\nu_{1}+x_{2})^{2}(\nu_{1}-x_{1})^{2}(\nu_{1}+x_{1})^{2}p_{1}^{2}+x_{1}x_{2}(x_{1}+x_{2})(\nu_{1}-x_{2})(\nu_{1}+x_{2})^{2}$
$\displaystyle\hphantom{T_{11}=}{}\times(\nu_{1}-x_{1})(\nu_{1}+x_{1})^{2}p_{2}p_{1}-x_{1}x_{2}^{2}(\nu_{1}-x_{2})^{2}(\nu_{1}+x_{2})^{2}(\nu_{1}+x_{1})^{2}p_{2}^{2}$
$\displaystyle\hphantom{T_{11}=}{}-2x_{1}(-x_{2}+x_{1})(\nu_{1}+x_{2})^{2}(\nu_{1}-x_{1})(\nu_{1}+x_{1})^{2}(c_{11}\nu_{1}+x_{2}c_{22})p_{1}+2x_{2}(-x_{2}+x_{1})$
$\displaystyle\hphantom{T_{11}=}{}\times(\nu_{1}-x_{2})(\nu_{1}+x_{2})^{2}(\nu_{1}+x_{1})^{2}(c_{11}\nu_{1}+x_{1}c_{22})p_{2}+4x_{1}x_{2}\nu_{1}^{3}(-x_{2}+x_{1})^{2}I_{22}$
$\displaystyle\hphantom{T_{11}=}{}-4c_{11}\nu_{1}c_{22}(-x_{2}+x_{1})^{2}(\nu_{1}+x_{2})^{2}(\nu_{1}+x_{1})^{2}\bigr{)}\bigl{(}\big{(}x_{1}x_{2}(\nu_{1}-x_{1})(\nu_{1}+x_{1})p_{1}$
$\displaystyle\hphantom{T_{11}=}{}-x_{1}x_{2}(\nu_{1}-x_{2})(\nu_{1}+x_{2})p_{2}+2(-x_{2}+x_{1})\big{(}c_{11}\nu_{1}^{2}x_{1}c_{22}x_{2}\big{)}\big{)}(-x_{2}+x_{1})(\nu_{1}+x_{2})$
$\displaystyle\hphantom{T_{11}=}{}\times(\nu_{1}+x_{1})\bigr{)}^{-1},$
$\displaystyle
T_{12}=\frac{x_{1}x_{2}}{2k\nu_{1}}\bigl{(}x_{1}^{2}(\nu_{1}+x_{2})^{2}(\nu_{1}-x_{1})^{2}(\nu_{1}+x_{1})^{2}p_{1}^{2}-2x_{1}x_{2}(\nu_{1}-x_{2})(\nu_{1}+x_{2})^{2}(\nu_{1}-x_{1})$
$\displaystyle\hphantom{T_{12}=}{}\times(\nu_{1}+x_{1})^{2}p_{2}p_{1}+x_{2}^{2}(\nu_{1}-x_{2})^{2}(\nu_{1}+x_{2})^{2}(\nu_{1}+x_{1})^{2}p_{2}^{2}+4x_{1}c_{22}(-x_{2}+x_{1})$
$\displaystyle\hphantom{T_{12}=}{}\times(\nu_{1}+x_{2})^{2}(\nu_{1}-x_{1})(\nu_{1}+x_{1})^{2}p_{1}-4x_{2}c_{22}(-x_{2}+x_{1})(\nu_{1}-x_{2})(\nu_{1}+x_{2})^{2}(\nu_{1}+x_{1})^{2}$
$\displaystyle\hphantom{T_{12}=}{}\times
p_{2}+4\nu_{1}^{4}(-x_{2}+x_{1})^{2}I_{22}+4c_{22}^{2}(-x_{2}+x_{1})^{2}(\nu_{1}+x_{2})^{2}(\nu_{1}+x_{1})^{2}\bigr{)}\bigl{(}(x_{1}x_{2}(\nu_{1}^{2}-x_{1}^{2})p_{1}$
$\displaystyle\hphantom{T_{12}=}{}\times-
x_{1}x_{2}(\nu_{1}^{2}-x_{2}^{2})p_{2}+2(x_{1}-x_{2})(c_{11}\nu_{1}^{2}+x_{1}c_{22}x_{2}))(x_{1}-x_{2})(\nu_{1}+x_{2})(\nu_{1}+x_{1})\bigr{)}^{-1},$
$\displaystyle
T_{21}=-\frac{k}{2\nu_{1}x_{1}x_{2}}\bigl{(}x_{1}^{2}x_{2}^{2}(\nu_{1}+x_{2})^{2}(\nu_{1}-x_{1})^{2}(\nu_{1}+x_{1})^{2}p_{1}^{2}-2x_{1}^{2}x_{2}^{2}(\nu_{1}-x_{2})(\nu_{1}+x_{2})^{2}$
$\displaystyle\hphantom{T_{21}=}{}\times(\nu_{1}-x_{1})(\nu_{1}+x_{1})^{2}p_{2}p_{1}+x_{1}^{2}x_{2}^{2}(\nu_{1}-x_{2})^{2}(\nu_{1}+x_{2})^{2}(\nu_{1}+x_{1})^{2}p_{2}^{2}+4c_{11}\nu_{1}x_{1}x_{2}$
$\displaystyle\hphantom{T_{21}=}{}\times(-x_{2}+x_{1})(\nu_{1}+x_{2})^{2}(\nu_{1}-x_{1})(\nu_{1}+x_{1})^{2}p_{1}-4c_{11}\nu_{1}x_{1}x_{2}(-x_{2}+x_{1})(\nu_{1}-x_{2})$
$\displaystyle\hphantom{T_{21}=}{}\times(\nu_{1}+x_{2})^{2}(\nu_{1}+x_{1})^{2}p_{2}+4x_{1}^{2}\nu_{1}^{2}x_{2}^{2}(-x_{2}+x_{1})^{2}I_{22}+4\nu_{1}^{2}c_{11}^{2}(-x_{2}+x_{1})^{2}(\nu_{1}+x_{2})^{2}$
$\displaystyle\hphantom{T_{21}=}{}\times(\nu_{1}+x_{1})^{2}\bigr{)}\bigl{(}\big{(}x_{1}x_{2}(x_{1}-x_{2})(\nu_{1}+x_{2})(\nu_{1}-x_{1})(\nu_{1}+x_{1})^{2}p_{1}-x_{1}x_{2}(x_{1}-x_{2})$
$\displaystyle\hphantom{T_{21}=}{}\times(\nu_{1}-x_{2})(\nu_{1}+x_{2})^{2}(\nu_{1}+x_{1})p_{2}+2(x_{1}-x_{2})^{2}(\nu_{1}+x_{2})(\nu_{1}+x_{1})$
$\displaystyle\hphantom{T_{21}=}{}\times\big{(}c_{11}\nu_{1}^{2}+x_{1}c_{22}x_{2}\big{)}\big{)}\bigr{)}^{-1},$
$\displaystyle T_{22}=-T_{11},$

where for the purpose of simplicity we have put $I_{11}=0$, $I_{12}=0$.

##  Appendix B The reconstruction formulae for the second SoV

Let us now reconstruct the dynamical variables $S_{ij}$, $T_{ij}$, $i,j=1,2$
using the variables of separation $p_{i}$, $x_{j}$ constructed in Section 5.3
and the Casimir functions. For this purpose we solve a system of eight linear-
quadratic equations in eight variables $S_{ij}$, $T_{ij}$:

$\displaystyle(x_{1}+x_{2})=-\frac{\nu_{1}}{k}(2k^{2}S_{12}+2kT_{11}+2kS_{22}+2T_{21}-2k^{2}T_{12}-S_{21}-2kS_{11}-2kT_{22})$
$\displaystyle\hphantom{(x_{1}+x_{2})=}{}\times\bigl{(}(2c_{1}-1)S_{11}+2kS_{12}+k(2c_{1}+1)S_{22}+(2c_{1}-1)T_{11}+2kT_{12}$
$\displaystyle\hphantom{(x_{1}+x_{2})=}{}+(2c_{1}+1)T_{22}+4(c_{11}-c_{22})\bigr{)}^{-1},$
(B.1a) $\displaystyle
x_{1}x_{2}=\frac{\nu_{1}^{2}}{k}(k(2c_{1}+1)S_{11}-2S_{21}-k(2c_{1}-1)S_{22}-k(2c_{1}+1)T_{11}-2T_{21}-k(2c_{1}-1)T_{22}$
$\displaystyle\hphantom{x_{1}x_{2}=}{}-4k(c_{11}-c_{22})\bigl{(}(2c_{1}-1)S_{11}+2kS_{12}+k(2c_{1}+1)S_{22}+(2c_{1}-1)T_{11}+2kT_{12}$
$\displaystyle\hphantom{x_{1}x_{2}=}{}+(2c_{1}+1)T_{22}+4(c_{11}-c_{22})\bigr{)}^{-1},$
(B.1b)
$\displaystyle-4\big{(}x_{1}^{2}-\nu_{1}^{2}\big{)}x_{1}p_{1}=\bigl{(}((3-2c_{1})S_{11}-4S_{12}k+(-2c_{1}-1)S_{22}+(3-2c_{1})T_{11}-4T_{12}k$
$\displaystyle\hphantom{-4\big{(}x_{1}^{2}-\nu_{1}^{2}\big{)}x_{1}p_{1}=}{}+(-2c_{1}-1)T_{22}-8c_{11})x_{1}^{2}+(4S_{11}\nu_{1}-4T_{11}\nu_{1}-4kS_{12}\nu_{1}$
$\displaystyle\hphantom{-4\big{(}x_{1}^{2}-\nu_{1}^{2}\big{)}x_{1}p_{1}=}{}+4kT_{12}\nu_{1})x_{1}+\nu_{1}^{2}(2c_{1}+1)S_{11}+\nu_{1}^{2}(2c_{1}+1)S_{22}+\nu_{1}^{2}(2c_{1}+1)T_{11}$
$\displaystyle\hphantom{-4\big{(}x_{1}^{2}-\nu_{1}^{2}\big{)}x_{1}p_{1}=}{}+\nu_{1}^{2}(2c_{1}+1)T_{22}+8c_{11}\nu_{1}^{2}\bigr{)},$
(B.1c)
$\displaystyle-4\big{(}x_{2}^{2}-\nu_{1}^{2}\big{)}x_{2}p_{2}=\bigl{(}((3-2c_{1})S_{11}-4S_{12}k+(-2c_{1}-1)S_{22}+(3-2c_{1})T_{11}-4T_{12}k$
$\displaystyle\hphantom{-4\big{(}x_{2}^{2}-\nu_{1}^{2}\big{)}x_{2}p_{2}=}{}+(-2c_{1}-1)T_{22}-8c_{11})x_{2}^{2}+(4S_{11}\nu_{1}-4T_{11}\nu_{1}-4kS_{12}\nu_{1}$
$\displaystyle\hphantom{-4\big{(}x_{2}^{2}-\nu_{1}^{2}\big{)}x_{2}p_{2}=}{}+4kT_{12}\nu_{1})x_{2}+\nu_{1}^{2}(2c_{1}+1)S_{11}+\nu_{1}^{2}(2c_{1}+1)S_{22}+\nu_{1}^{2}(2c_{1}+1)T_{11}$
$\displaystyle\hphantom{-4\big{(}x_{2}^{2}-\nu_{1}^{2}\big{)}x_{2}p_{2}=}{}+\nu_{1}^{2}(2c_{1}+1)T_{22}+8c_{11}\nu_{1}^{2}\bigr{)},$
(B.1d) $\displaystyle I_{21}=S_{11}S_{22}-S_{12}S_{21},$ (B.1e) $\displaystyle
I_{22}=T_{11}T_{22}-T_{12}T_{21},$ (B.1f) $\displaystyle
I_{11}=S_{11}+S_{22},$ (B.1g) $\displaystyle I_{12}=T_{11}+T_{22}.$ (B.1h)

The following proposition is proven by the direct calculations.

######  Proposition B.1.

The system of equations (B.1) is solved with respect of the variables
$S_{ij}$, $T_{ij}$, $i,j=1,2$ as follows

$\displaystyle
S_{11}=-\frac{1}{2\nu_{1}}\bigl{(}x_{1}^{2}x_{2}(-x_{2}+\nu_{1})^{2}\big{(}\nu_{1}^{2}-x_{1}^{2}\big{)}^{2}p_{1}^{2}-x_{2}x_{1}(x_{1}+x_{2})(x_{2}+\nu_{1})(\nu_{1}-x_{2})^{2}$
$\displaystyle\hphantom{S_{11}=}{}\times\big{(}\nu_{1}^{2}-x_{1}^{2}\big{)}^{2}p_{2}p_{1}+x_{2}^{2}x_{1}(-x_{2}+\nu_{1})^{2}(x_{2}+\nu_{1})^{2}(\nu_{1}-x_{1})^{2}p_{2}^{2}-2x_{1}(-x_{2}+x_{1})$
$\displaystyle\hphantom{S_{11}=}{}\times(-x_{2}+\nu_{1})^{2}(x_{1}+\nu_{1})(-x_{1}+\nu_{1})^{2}(c_{11}\nu_{1}-c_{22}x_{2}))p_{1}+2x_{2}(x_{1}-x_{2})(x_{2}+\nu_{1})$
$\displaystyle\hphantom{S_{11}=}{}\times(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}(c_{11}\nu_{1}-c_{22}x_{1})p_{2}+4x_{1}\nu_{1}^{3}x_{2}(x_{1}-x_{2})^{2}I_{21}-4c_{11}\nu_{1}c_{22}$
$\displaystyle\hphantom{S_{11}=}{}\times(x_{1}-x_{2})^{2}(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}\bigr{)}\bigl{(}\big{(}{-}x_{1}x_{2}(x_{1}+\nu_{1})(\nu_{1}-x_{1})p_{1}+x_{1}x_{2}(x_{2}+\nu_{1})$
$\displaystyle\hphantom{S_{11}=}{}\times(\nu_{1}-x_{2})p_{2}+2(x_{1}-x_{2})\big{(}x_{1}c_{22}x_{2}+c_{11}\nu_{1}^{2}\big{)}\big{)}(x_{1}-x_{2})(\nu_{1}-x_{2})(\nu_{1}-x_{1})\bigr{)}^{-1},$
$\displaystyle
S_{12}=-\frac{1}{2kx_{1}x_{2}}\bigl{(}x_{2}^{2}x_{1}^{2}(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}(x_{1}+\nu_{1})^{2}p_{1}^{2}-2x_{2}^{2}x_{1}^{2}(x_{2}+\nu_{1})(\nu_{1}-x_{2})^{2}$
$\displaystyle\hphantom{S_{12}=}{}\times(x_{1}+\nu_{1})(\nu_{1}-x_{1})^{2}p_{2}p_{1}+x_{2}^{2}x_{1}^{2}(\nu_{1}-x_{2})^{2}(x_{2}+\nu_{1})^{2}(\nu_{1}-x_{1})^{2}p_{2}^{2}-4\nu_{1}x_{2}c_{11}x_{1}$
$\displaystyle\hphantom{S_{12}=}{}\times(x_{1}-x_{2})(\nu_{1}-x_{2})^{2}(x_{1}+\nu_{1})(\nu_{1}-x_{1})^{2}p_{1}+4\nu_{1}x_{2}c_{11}x_{1}(x_{1}-x_{2})(x_{2}+\nu_{1})$
$\displaystyle\hphantom{S_{12}=}{}\times(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}p_{2}+4x_{2}^{2}\nu_{1}^{2}x_{1}^{2}(x_{1}-x_{2})^{2}I_{21}+4\nu_{1}^{2}c_{11}^{2}(x_{1}-x_{2})^{2}(\nu_{1}-x_{2})^{2}$
$\displaystyle\hphantom{S_{12}=}{}\times(\nu_{1}-x_{1})^{2}\bigr{)}\bigl{(}\big{(}{-}x_{2}x_{1}(x_{1}-x_{2})(\nu_{1}-x_{2})(x_{1}+\nu_{1})(\nu_{1}-x_{1})^{2}p_{1}+x_{2}x_{1}(x_{1}-x_{2})$
$\displaystyle\hphantom{S_{12}=}{}\times(x_{2}+\nu_{1})(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})p_{2}+2(x_{1}-x_{2})^{2}(\nu_{1}-x_{2})(\nu_{1}-x_{1})$
$\displaystyle\hphantom{S_{12}=}{}\times\big{(}x_{1}c_{22}x_{2}+c_{11}\nu_{1}^{2}\big{)}\big{)}\bigr{)}^{-1},$
$\displaystyle
S_{21}=\frac{x_{2}x_{1}k}{2\nu_{1}^{2}}\bigl{(}x_{1}^{2}(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}(x_{1}+\nu_{1})^{2}p_{1}^{2}-2x_{1}x_{2}(x_{2}+\nu_{1})(\nu_{1}-x_{2})^{2}(x_{1}+\nu_{1})$
$\displaystyle\hphantom{S_{21}=}{}\times(\nu_{1}-x_{1})^{2}p_{2}p_{1}+x_{2}^{2}(\nu_{1}-x_{2})^{2}(x_{2}+\nu_{1})^{2}(\nu_{1}-x_{1})^{2}p_{2}^{2}+4c_{22}x_{1}(x_{1}-x_{2})(\nu_{1}-x_{2})^{2}$
$\displaystyle\hphantom{S_{21}=}{}\times(x_{1}+\nu_{1})(\nu_{1}-x_{1})^{2}p_{1}-4c_{22}x_{2}(x_{1}-x_{2})(x_{2}+\nu_{1})(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}p_{2}$
$\displaystyle\hphantom{S_{21}=}{}+4\nu_{1}^{4}(x_{1}-x_{2})^{2}I_{21}+4c_{22}^{2}(x_{1}-x_{2})^{2}(\nu_{1}-x_{2})^{2}(\nu_{1}-x_{1})^{2}\bigr{)}\bigl{(}\big{(}{-}x_{1}x_{2}(\nu_{1}-x_{1})$
$\displaystyle\hphantom{S_{21}=}{}\times(x_{1}+\nu_{1})(x_{1}-x_{2})p_{1}+x_{1}x_{2}(\nu_{1}-x_{2})(x_{2}+\nu_{1})(x_{1}-x_{2})p_{2}+2(x_{1}-x_{2})^{2}$
$\displaystyle\hphantom{S_{21}=}{}\times\big{(}x_{1}c_{22}x_{2}+c_{11}\nu_{1}^{2}\big{)}\big{)}(\nu_{1}-x_{2})(\nu_{1}-x_{1})\bigr{)}^{-1},$
$\displaystyle S_{22}=-S_{11},$ $\displaystyle
T_{11}=-\frac{1}{2\nu_{1}}\bigl{(}-x_{1}^{2}x_{2}(x_{2}+\nu_{1})^{2}\big{(}\nu_{1}^{2}-x_{1}^{2}\big{)}^{2}p_{1}^{2}+x_{1}x_{2}(x_{1}+x_{2})(\nu_{1}-x_{2})(x_{2}+\nu_{1})^{2}(\nu_{1}-x_{1})$
$\displaystyle\hphantom{T_{11}=}{}\times(x_{1}+\nu_{1})^{2}p_{2}p_{1}-x_{2}^{2}x_{1}(\nu_{1}-x_{2})^{2}(x_{2}+\nu_{1})^{2}(x_{1}+\nu_{1})^{2}p_{2}^{2}+2x_{1}(x_{1}-x_{2})$
$\displaystyle\hphantom{T_{11}=}{}\times(x_{2}+\nu_{1})^{2}(\nu_{1}-x_{1})(x_{1}+\nu_{1})^{2}(c_{11}\nu_{1}+c_{22}x_{2}))p_{1}-2x_{2}(x_{1}-x_{2})(\nu_{1}-x_{2})$
$\displaystyle\hphantom{T_{11}=}{}\times(x_{2}+\nu_{1})^{2}(x_{1}+\nu_{1})^{2}(c_{11}\nu_{1}+c_{22}x_{1})p_{2}+4x_{1}\nu_{1}^{3}x_{2}(x_{1}-x_{2})^{2}I_{22}-4c_{11}\nu_{1}c_{22}$
$\displaystyle\hphantom{T_{11}=}{}\times(x_{1}-x_{2})^{2}(x_{2}+\nu_{1})^{2}(x_{1}+\nu_{1})^{2}\bigr{)}\bigl{(}(x_{1}-x_{2})\big{(}{-}x_{1}x_{2}(\nu_{1}-x_{1})(x_{1}+\nu_{1})^{2}(x_{2}+\nu_{1})p_{1}$
$\displaystyle\hphantom{T_{11}=}{}+x_{1}x_{2}(\nu_{1}-x_{2})(x_{2}+\nu_{1})^{2}(x_{1}+\nu_{1})p_{2}+2(x_{1}-x_{2})(x_{2}+\nu_{1})(x_{1}+\nu_{1})$
$\displaystyle\hphantom{T_{11}=}{}\times\big{(}x_{1}c_{22}x_{2}+c_{11}\nu_{1}^{2}\big{)}\big{)}\bigr{)}^{-1},$
$\displaystyle
T_{12}=-\frac{x_{1}x_{2}k}{2}\bigl{(}x_{1}^{2}x_{2}^{2}(x_{2}+\nu_{1})^{2}\big{(}\nu_{1}^{2}-x_{1}^{2}\big{)}^{2}p_{1}^{2}-2x_{1}^{2}x_{2}^{2}(\nu_{1}-x_{2})(x_{2}+\nu_{1})^{2}(\nu_{1}-x_{1})$
$\displaystyle\hphantom{T_{12}=}{}\times(x_{1}+\nu_{1})^{2}p_{2}p_{1}+x_{1}^{2}x_{2}^{2}(\nu_{1}-x_{2})^{2}(x_{2}+\nu_{1})^{2}(x_{1}+\nu_{1})^{2}p_{2}^{2}-4\nu_{1}x_{2}c_{11}x_{1}(x_{1}-x_{2})$
$\displaystyle\hphantom{T_{12}=}{}\times(x_{2}+\nu_{1})^{2}(\nu_{1}-x_{1})(x_{1}+\nu_{1})^{2}p_{1}+4\nu_{1}x_{2}c_{11}x_{1}(x_{1}-x_{2})(\nu_{1}-x_{2})(x_{2}+\nu_{1})^{2}$
$\displaystyle\hphantom{T_{12}=}{}\times(x_{1}+\nu_{1})^{2}p_{2}+4\nu_{1}^{2}x_{2}^{2}x_{1}^{2}(x_{1}-x_{2})^{2}I_{22}+4\nu_{1}^{2}c_{11}^{2}(x_{1}-x_{2})^{2}(x_{2}+\nu_{1})^{2}(x_{1}+\nu_{1})^{2}\bigr{)}$
$\displaystyle\hphantom{T_{12}=}{}\times\bigl{(}\big{(}{-}x_{1}x_{2}(x_{1}-x_{2})(x_{2}+\nu_{1})(\nu_{1}-x_{1})(x_{1}+\nu_{1})^{2}p_{1}+x_{1}x_{2}(x_{1}-x_{2})(\nu_{1}-x_{2})$
$\displaystyle\hphantom{T_{12}=}{}\times(x_{2}+\nu_{1})^{2}(x_{1}+\nu_{1})p_{2}+2(x_{1}-x_{2})^{2}(x_{2}+\nu_{1})(x_{1}+\nu_{1})\big{(}x_{1}c_{22}x_{2}+c_{11}\nu_{1}^{2}\big{)}\big{)}\bigr{)}^{-1},$
$\displaystyle
T_{21}=\frac{kx_{1}x_{2}}{2\nu_{1}^{2}}\bigl{(}x_{1}^{2}(x_{2}+\nu_{1})^{2}\big{(}\nu_{1}^{2}-x_{1}^{2}\big{)}^{2}p_{1}^{2}-2x_{1}x_{2}(\nu_{1}-x_{2})(x_{2}+\nu_{1})^{2}(\nu_{1}-x_{1})$
$\displaystyle\hphantom{T_{21}=}{}\times(x_{1}+\nu_{1})^{2}p_{2}p_{1}+x_{2}^{2}(\nu_{1}-x_{2})^{2}(x_{2}+\nu_{1})^{2}(x_{1}+\nu_{1})^{2}p_{2}^{2}-4c_{22}x_{1}(x_{1}-x_{2})$
$\displaystyle\hphantom{T_{21}=}{}\times(x_{2}+\nu_{1})^{2}(\nu_{1}-x_{1})(x_{1}+\nu_{1})^{2}p_{1}+4c_{22}x_{2}(x_{1}-x_{2})(\nu_{1}-x_{2})(x_{2}+\nu_{1})^{2}$
$\displaystyle\hphantom{T_{21}=}{}\times(x_{1}+\nu_{1})^{2}p_{2}+4\nu_{1}^{4}(x_{1}-x_{2})^{2}I_{22}+4c_{22}^{2}(x_{1}-x_{2})^{2}(x_{2}+\nu_{1})^{2}(x_{1}+\nu_{1})^{2}\bigr{)}$
$\displaystyle\hphantom{T_{21}=}{}\times\bigl{(}(-x_{1}x_{2}(\nu_{1}-x_{1})(x_{1}+\nu_{1})(x_{1}-x_{2})p_{1}+x_{1}x_{2}(\nu_{1}-x_{2})(x_{2}+\nu_{1})(x_{1}-x_{2})p_{2}$
$\displaystyle\hphantom{T_{21}=}{}+2(x_{1}-x_{2})^{2}(x_{1}c_{22}x_{2}+c_{11}\nu_{1}^{2}))(x_{2}+\nu_{1})(x_{1}+\nu_{1})\bigr{)}^{-1},$
$\displaystyle T_{22}=-T_{11},$

where for the purpose of simplicity we have put $I_{11}=0$, $I_{12}=0$.

### Acknowledgements

The author is grateful to the late Boris Dubrovin for the discussions. The
work over this paper was partially supported by the Division of Physics and
Astronomy of NAS of Ukraine (Project No. 0117U000240).

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