# Evaluation of Social Biases in Recent Large Pre-Trained Models

Swapnil Sharma &Nikita Anand &Kranthi Kiran GV &Alind Jain

###### Abstract

Large pre-trained language models are widely used in the community. These
models are usually trained on unmoderated and unfiltered data from open
sources like the internet. Due to this, biases that we see in platforms online
which are a reflection of those in society are in turn captured and learnt by
these models. These models are deployed in applications that affect millions
of people and their inherent biases are harmful to the targeted social groups.
In this work, we study the general trend in bias reduction as newer pre-
trained models are released. Three recent models ( ELECTRA, DeBERTa and
DistilBERT) are chosen and evaluated against two bias benchmarks, StereoSet
and CrowS-Pairs. They are compared to the baseline of BERT using the
associated metrics. We explore whether as advancements are made and newer,
faster, lighter models are released: are they being developed responsibly such
that their inherent social biases have been reduced compared to their older
counterparts? The results are compiled and we find that all the models under
study do exhibit biases but have generally improved as compared to BERT. Link
to the code is available on Github.111 https://github.com/KranthiGV/bias-
evaluation

##  1 Introduction

Given the popularity of large pre-trained language models and their widespread
use in the community, it is pertinent to evaluate their inherent biases. These
biases come about because these models (for example, BERT (Devlin et al.,
2019) and BERT based models, ALBERT (Lan et al., 2020), RoBERTa (Liu et al.,
2019)) are trained on large sets of often freely available and unmoderated
text data from sources such as the internet.

Due to this, gender, race, religious and other social biases that we see in
the real world are often translated into the models. Since these models are
deployed in applications that are used by a large number of people, this bias
is harmful. For example, these models are commonly used in content moderation
tasks on social media platforms online. If the model is unfairly biased
against certain social groups- minorities, marginalized people and those
sections of society that are discriminated against, then they would be
adversely affected by the application. So it is of utmost importance to ensure
that models are trained in such a way that the bias is mitigated or to de-bias
them.

There are multiple studies (Meade et al., 2022; Ahn and Oh, 2021; Bhardwaj et
al., 2021; Kurita et al., 2019) on social biases in BERT and older language
models. We extend this and evaluate the relatively newer models: ELECTRA
(Clark et al., 2020), DistilBERT (Sanh et al., 2019) and DeBERTa (He et al.,
2020). These models have gained popularity due to factors such as better
performance, low computer requirements and parameter efficiency. Each model is
evaluated against two bias evaluation datasets, StereoSet (Nadeem et al.,
2021) and Crowdsourced Stereotype Pairs, CrowS-Pairs (Nangia et al., 2020).
StereoSet accounts for four different biases that are based on- gender,
occupation, race and religion. In addition to these, CrowS-Pairs also has
sexual orientation, age, nationality, disability and physical appearance.

We aim to answer the question: as technology advances and newer pre-trained
language models are released, are they being done responsibly such that their
inherent social biases have been reduced? We believe that while making
improvements to speed, size, efficiency and performance of models is at the
forefront of research, equal importance needs to be given to mitigating their
biases given their impact in the real world.

The three newer models are run against these benchmarks and compared to the
baseline of BERT. This gives us a general trend in whether bias is reduced as
advancements in language models are being made. We find that all the models
exhibit bias, but the three recent models perform better on the benchmarks
than BERT. Models have varying results across different bias categories and we
attempt to analyse their behaviour.

##  2 Related work

The empirical survey conducted in Meade et al. (2022) studied five different
debiasing techniques Counterfactual Data Augmentation (Zmigrod et al., 2019),
Dropout (Beutel et al., 2020), Iterative Nullspace Projection (Ravfogel et
al., 2020), Self-Debias (Schick et al., 2021) and SentenceDebias (Liang et
al., 2020) on racial, religious and gender biases. They applied these
techniques on BERT, RobERTa, ALBERT and GPT-2 (Radford et al., 2019) and
evaluated them against three bias benchmarks (StereoSet, CrowS-Pairs and
Sentence Encoder Association Test (May et al., 2019). They report which
technique is best at debiasing and whether these techniques hamper a model’s
language modeling ability and performance on downstream Natural Language
Understanding tasks.

Crowdsourced Stereotype Pairs, CrowS-Pairs (Nangia et al., 2020) is a
crowdsourced bias evaluation dataset that deals with nine different types of
social bias. Each sample consists of a pair of sentences- a stereotype and an
anti-stereotype. This high quality dataset provides us a good way to gain
insights on varying types of bias in recent models.

An earlier bias evaluation benchmark StereoSet (Nadeem et al., 2021) deals
with four types of biases but on a larger scale. They introduce the Context
Association Test that "measures the language modeling ability as well as the
stereotypical bias of pretrained language models". This work provides us with
a large amount of data, though it does not cover as many biases as CrowS-
Pairs. StereoSet has two kinds of tasks, intrasentence, which is sentence
level and intersentence, which is discourse level. The intrasentence task has
a fill-in-the-blank style structure describing a group where there is three
options for the attribute, a stereotype, an anti-stereotype and an unrelated
one. The intersentence task gives a context about the target group and the
second contains the attribute. CrowS-Pairs and StereoSet’s intrasentence task
both have the similar structure of two sentences, the first being more biased
that the second. The difference is that in the former the group being
discussed changes between the two sentences while in the latter, the attribute
assigned to the group changes. CrowS-Pairs conducted a validation study and
came out rated much higher at 80% compared to StereoSet’s 62%. Since CrowS-
Pairs improves only on StereoSet’s intrasentence task, we use it alongside the
intersentence task.

Bhardwaj et al. (2021) does a deep-dive into gender bias in BERT. They examine
the bias brought about by BERT in five downstream tasks based on sentiment and
emotion intensity in Tweets. It is pointed out that intuitively and ideally
the words that are gender indicative should matter little for the model’s
predictions but in actuality, they turn out to influence the predictions
heavily. A layer-level solution is proposed wherein directions that encode
gender are picked out and embeddings in the same are removed to largely reduce
bias in the downstream tasks. This is one of the first studies that analyses
bias in BERT in this manner.

A different perspective to the bias problem is offered by Ahn and Oh (2021)
where the language dependent nature of bias is reported. Ethnic bias is
studied across multiple languages using various monolingual versions of BERT.
They alleviate bias using two methods: using a multilingual model and aligning
words in two multilingual models.

##  3 Methodology

The methodology consists of the following stages: (1) Data loading and
preprocessing, (2) Model loading and (3) Evaluation. We replicate the
evaluation pipeline in CrowS-Pairs and StereoSet.

In data loading and preprocessing stage, we load a particular bias benchmark
dataset and process it to ingest it further into the pipeline. This includes
tokenization according the pre-trained model chosen. Evaluation consists of
computing the metrics after running the given model on the bias benchmark
dataset chosen.

###  3.1 Choice of Models

Our choice of recent pre-trained models largely depends on the Hugging Face
(Wolf et al., 2020) downloads and citation count as a guide to how popular and
used they are. We compare them to a BERT baseline.

##### DistilBERT

retains 97% of BERT’s language understanding capabilities while being 40%
smaller and 60% faster. This model has been downloaded from the Hugging Face
library over 5 million times in the last month alone and has about 1700
citations.

##### DeBERTa

consistently outperforms a BERT based model RoBERTa on many NLP tasks. It has
been downloaded over 1 million times over the last month which indicates its
popularity in the community.

##### ELECTRA

shows significant improvement in contextual representations over BERT given
the same data and compute using a novel pre-training technique. ELECTRA
(small, base and large) had over 600,000 downloads and 1270 citations.

###  3.2 Datasets

####  3.2.1 StereoSet

Stereoset dataset contains about 16,995 test sentences evaluating biases in
four domains which are race, profession, gender and religion. Nearly 85% of
the examples are of profession or race bias categories and religion
constitutes merely 4% of the total.

The dataset contains two kinds of context association tests (CAT):
Intersentence (sentence level) and Intrasentence (discourse level), both of
which contain a Context, a Stereotype, an Anti-Stereotype and an Unrelated
component.

The evaluation metrics defined in Nadeem et al. (2021) are Language Modeling
Score (lms), Stereotype Score (ss) and a combination of the two, Idealized CAT
Score (icat). As per the paper, lms of a target term is "the percentage of
instances in which a language model prefers the meaningful over meaningless
association" and ss is "the percentage of examples in which a model prefers a
stereotypical association over an anti-stereotypical association". The
combination of the two gives form to icat which is mathematically described as
follows,

$\displaystyle icat=\frac{lms\times min(ss,100-ss)}{50}$ (1)

This metric captures the ability of a model to remain unbiased while
performing language modeling extremely well.

####  3.2.2 CrowS-Pairs

CrowS-Pairs is a well curated, crowdsourced dataset that covers nine types of
social biases. Each example comes in pairs of a stereotype and a corresponding
anti-stereotype. To do this, crowdworkers were asked to write a stereotyping
statement about a targeted group and then make minimal changes so that the
target group is now an advantaged group instead. In this process, the words
that do not change are called unmodified tokens and the ones that do are
called modified tokens. They performed a series of checks to ensure the data
is validated. The resulting dataset has 1508 examples, with the race category
comprising nearly third of the total. 73.5% of the data belongs to either the
race, gender, occupation or nationality categories. They introduce their own
metric, given by the _pseudo-log-likelihood_ scoring,

$\displaystyle score(S)=\sum_{i=0}^{|C|}\log P(u_{i}\in U|U_{\char 92\relax
u_{i}},M,\theta)$ (2)

In this equation, U is the set of unmodified tokens, M is the set of modified
tokens and S is the union of the two.

| BERT${}_{\emph{base}}(baseline)$ | DistilBERT${}_{\emph{base}}$ | DeBERTa${}_{\emph{base}}$ | ELECTRA${}_{\emph{base}}$
---|---|---|---|---
CrowS-Pairs | 60.48 | 56.83 | 52.59 | 56.76
CrowS-Pairs-_stereo_ | 61.09 | 57.33 | 53.85 | 56.98
CrowS-Pairs-_antistereo_ | 56.88 | 54.13 | 45.87 | 55.76
_Bias Categories in CrowS-Pairs_ |
Race / Color | 58.14 | 53.29 | 56.01 | 57.95
Gender / Gender Identity | 58.02 | 54.58 | 43.51 | 52.67
Socioeconomic status / Occupation | 59.88 | 55.81 | 55.23 | 61.63
Nationality | 62.89 | 54.09 | 55.97 | 53.46
Religion | 71.43 | 70.48 | 49.52 | 65.71
Age | 55.17 | 59.77 | 48.28 | 56.32
Sexual Orientation | 67.86 | 70.24 | 59.52 | 50.0
Physical Appearance | 63.49 | 55.56 | 53.97 | 53.97
Disability | 61.67 | 61.67 | 46.67 | 56.67
Table 1: Model performances on CrowS-Pairs. Higher numbers indicate higher bias. Low bias | He could not believe his wife could be so callous.
---|---
He could not believe his husband could be so callous.
High bias | Gay people are really flamboyant and are always wearing ridiculous costumes.
Straight people are really flamboyant and are always wearing ridiculous
costumes.
Table 2: Examples of sentence pairs in CrowS-Pairs on which ELECTRA exhibited
low bias and high bias.

##  4 Experiments

In this section, we describe the process followed to evaluate the models and
the experimental results.

##### Stereoset Set Up

The task at hand in stereoset inter-sentences is to pick an attribute sentence
give a target term sentence. This task is very similar to the Next Sentence
Prediction (NSP) task that was one of the original BERT training tasks. To
better analyze the model bias on varied tasks we analyze them on Stereoset
inter-sentence dataset.

However, none of the models under consideration apart from BERT-baseline,
contain a pretrained NSP classification head. So we decided to train the NSP
heads on a subset of the Wikipedia corpus (Foundation, ) for these models. Our
NSP classification head achieves a 94.88% on DistilBERT-_base_ , 96.18% on
BERT-_base_ , 95.18% on DeBERTa-_base_ and 95.48% on ELECTRA-_base_.

Bias | Count | lms | ss | icat
---|---|---|---|---
BERT${}_{\emph{base}}(baseline)$
Gender | 726 | 89.26 | 59.09 | 73.03
Profession | 2481 | 84.70 | 63.48 | 61.86
Race | 2928 | 87.76 | 60.35 | 69.59
Religion | 234 | 91.03 | 61.54 | 70.02
Overall | 2123 | 86.86 | 61.47 | 66.99
DistilBERT${}_{\emph{base}}$
Gender | 726 | 87.56 | 56.78 | 75.70
Profession | 2481 | 86.50 | 58.25 | 72.24
Race | 2928 | 82.77 | 56.89 | 71.36
Religion | 234 | 86.29 | 62.02 | 65.55
Overall | 2123 | 84.92 | 57.59 | 72.04
DeBERTa${}_{\emph{base}}$
Gender | 726 | 77.49 | 8.46 | 13.11
Profession | 2481 | 80.25 | 24.03 | 38.57
Race | 2928 | 81.15 | 23.57 | 38.26
Religion | 234 | 82.49 | 18.69 | 30.83
Overall | 2123 | 80.40 | 21.65 | 34.81
ELECTRA${}_{\emph{base}}$
Gender | 726 | 90.79 | 56.60 | 78.81
Profession | 2481 | 87.98 | 60.60 | 69.33
Race | 2928 | 88.37 | 57.14 | 75.75
Religion | 234 | 97.52 | 53.84 | 90.02
Overall | 2123 | 88.87 | 58.26 | 74.19
Table 3: Model performances on StereoSet intersentences. Numbers greater than
50 indicate higher bias.

###  4.1 Results

In Table 1, we observe that on the CrowS-Pairs benchmark dataset, all the
models exhibit bias. BERT has the highest bias score while models such as
DeBERTa and ELECTRA perform better. DeBERTa has a metric score of $52.59$
which is the lowest score among the models we studied in this work. We also
observe that the models do not demonstrate similar scores in each of the bias
categories of CrowS-Pairs dataset. For instance, DistilBERT exhibits higher
bias on Religion ($70.48$) and Sexual Orientation ($70.24$) than other
categories such as Gender ($54.58$), Race ($53.29$) and others. ELECTRA,
however, exhibits lower bias on Religion ($65.71$) than DistilBERT.

Table 2 presents examples of sentences on which ELECTRA model exhibited low
and high bias. In the example on which it exhibited higher bias: "Gay people
are flamboyant…", it is likely due to the presence of words such as
"flamboyant" which may have been typically found along with the word "Gay" on
the training corpus. The examples on which it exhibited low bias, it could be
likely due to the words being quite general "wife" and "husband".

In Table 3, we observe the outcome running BERT against StereoSet. Similar to
CrowS-Pair evaluation, we observe that models have different scores in each
bias categories. The Language Modeling Score was slightly higher in the
Religion category as compared to the others.

DeBERTa in both CrowS-Pairs and stereoset shows inclination to anti-
stereotype. The anti-stereotype result seems more pronounced in case of
intersentences. We hypothesize that this effect is more vividly visible
because we have fine tuned DeBERTa for NSP task on Wikipedia Corpus. Since
DeBERTa has far less trainable parameters as compared to other models, it ends
up learning anti-stereotype behaviour from strictly monitored Wikipedia
content.

##  5 Ethical Considerations

The study we conduct in this work is meant to indicate the progress made in
mitigating social biases in recent models. A model performing well on these
bias benchmarks alone should not be interpreted as it being unbiased.

StereoSet only considers a binary definition of gender. Going forward we
believe it is crucial for studies on gender bias to be non-binary.

The benchmarks employed only North American crowdworkers so the stereotypes
are narrow in that regard and do not represent biases across cultures.

##  6 Collaboration Statement

Kranthi implemented the CrowS-Pairs evaluation and Swapnil worked on
StereoSet. Nikita worked on Introduction, Literature Review, Methodology and
experiments with SEAT (not included in this report). We worked together to
analyse the results. Alind worked on Datasets and everyone worked on the
presentation.

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##  Appendix A Appendix

In Table 4 we show the models evaluated on StereoSet Intrasentence dataset.
CrowS-Pairs found errors in the StereoSet Intrasentences and so we ask readers
to be cautious of these results.

Bias | Count | lms | ss | icat
---|---|---|---|---
BERT${}_{\emph{base}}(Baseline)$
Gender | 765 | 85.97 | 63.56 | 62.65
Profession | 2430 | 82.72 | 61.16 | 64.26
Race | 2886 | 85.45 | 57.24 | 73.07
Religion | 237 | 88.46 | 55.13 | 79.39
Overall | 2106 | 84.59 | 59.45 | 68.60
DistilBERT${}_{\emph{base}}$
Gender | 765 | 86.43 | 59.76 | 69.56
Profession | 2430 | 83.50 | 64.06 | 60.02
Race | 2886 | 86.71 | 58.05 | 72.74
Religion | 237 | 89.13 | 61.98 | 67.78
Overall | 2106 | 85.55 | 60.70 | 67.24
DeBERTa${}_{\emph{base}}$
Gender | 765 | 54.41 | 50.54 | 53.83
Profession | 2430 | 44.06 | 49.64 | 43.74
Race | 2886 | 49.02 | 50.64 | 48.40
Religion | 237 | 51.17 | 57.06 | 43.95
Overall | 2106 | 47.90 | 50.49 | 47.43
ELECTRA${}_{\emph{base}}$
Gender | 765 | 57.35 | 48.19 | 55.28
Profession | 2430 | 58.47 | 46.19 | 54.00
Race | 2886 | 59.09 | 43.83 | 51.80
Religion | 237 | 60.57 | 45.98 | 55.70
Overall | 2106 | 58.69 | 45.36 | 53.24
Table 4: StereoSet intrasentences results.


Non-Homogeneous Hydrodynamic Systems and Quasi-Stäckel Hamiltonians

Non-Homogeneous Hydrodynamic Systems
and Quasi-Stäckel Hamiltonians

Krzysztof MARCINIAK † and Maciej BŁASZAK ‡

K. Marciniak and M. Błaszak

† Department of Science and Technology, Campus Norrköping, Linköping
University, Sweden<EMAIL_ADDRESS>
‡ Faculty of Physics, Division of Mathematical Physics, A. Mickiewicz
University,
‡ Poznań, Poland<EMAIL_ADDRESS>
Received June 12, 2017, in final form September 25, 2017; Published online
September 28, 2017

In this paper we present a novel construction of non-homogeneous hydrodynamic
equations from what we call quasi-Stäckel systems, that is non-commutatively
integrable systems constructed from appropriate maximally superintegrable
Stäckel systems. We describe the relations between Poisson algebras generated
by quasi-Stäckel Hamiltonians and the corresponding Lie algebras of vector
fields of non-homogeneous hydrodynamic systems. We also apply Stäckel
transform to obtain new non-homogeneous equations of considered type.

Hamiltonian systems; superintegrable systems; Stäckel systems; hydrodynamic
systems; Stäckel transform

70H06; 70H20; 35F50; 53B20

##  1 Introduction

In the present paper we investigate a class of first-order quasi-linear PDEs
of the form

$\displaystyle q_{t}=K(q)q_{x}+Y(q),$ (1.1)

where $q=(q_{1},\dots,q_{n})^{T},$ $q_{i}=q_{i}(x,t)$, $K$ is an $n\times n$
matrix depending on $q$ and $Y$ is a vector depending on $q$ (vector field).
The systems (1.1) are called in literature non-homogenous hydrodynamic or
dispersionless systems. Specifically, we shall focus on the systems (1.1) with
the homogenous part $K(q)q_{x}$ being semi-Hamiltonian in the sense of Tsarev
[22] and weakly nonlinear [20]. The homogeneous case $Y=0$ has been
intensively studied in literature due to the fact that such equations are
strongly connected with classical integrable systems. For example, any
solution of the homogeneous equation $q_{t}=K(q)q_{x}$ is also a solution of
some particular Liouville integrable and separable finite-dimensional system,
so called Stäckel system [10, 12]. The matrices $K(q)$ can in this context be
interpreted as $(1,1)$-Killing tensors of Stäckel metrics [6].

The aim of this paper is to construct a class of non-homogeneous hydrodynamic
equations of type (1.1) that can still be related to finite-dimensional
integrable systems. This issue was first considered in [11, 13] but only for
the case of two integrals, where the authors searched, for a given quadratic
in momenta Hamiltonian, an integral of motion with quadratic in momenta terms
but also having additional terms linear in momenta (magnetic terms),
generating non-homogeneous terms in the related hydrodynamic equation. They
noticed that these linear in momenta terms are given by vectors that are
Killing vectors for the metric of the first Hamiltonian.

Another approach to the idea of constructing Liouville integrable systems with
linear in momenta terms (restricted to cases $n=2$ and $3$) has been presented
in [18] and later in [17], where the authors defined these systems as so
called quasi-Stäckel systems and also noted some connections of such systems
with non-homogenous hydrodynamic equations.

Inspired by these results we will look for a way – valid in any dimension – of
modifying the Stäckel Hamiltonians by linear in momenta terms (which makes the
corresponding hydrodynamic equations non-homogeneous) such that $(i)$ the
modified system defines a maximally superintegrable system where the extra
integrals of motion are all linear in momenta and $(ii)$ it constitutes a non-
commutative Poisson algebra. As a consequence, the structure constants of this
algebra and the algebra of corresponding non-homogeneous hydrodynamic vector
fields will be (up to a sign) the same. We call these modified systems quasi-
Stäckel systems as well, but we want to stress that our definition of quasi-
Stäckel systems is different from the definition in [17], as there the author
demands these systems to be Liouville integrable while our systems are
integrable only in the non-commutative sense.

The paper is organized as follows. In Section 2 we briefly remind the relation
between homogeneous hydrodynamic Killing equations and geodesic Stäckel
systems. In Section 3 we restrict our attention to a particular class of
Stäckel systems, namely Benenti class with constant curvature metrics (it
should be pointed out that there is a substantial literature on constant
curvature systems with magnetic terms, see for example [16] and references
there). Each system of this class consists of $n$ geodesic Hamiltonians
$E_{i}$. For this particular class we find $n-1$ Killing vectors $Y_{i}$ of
the metric tensor of the first Hamiltonian $E_{1},$ generating additional
linear in momenta integrals $W_{i}$ which yields a maximally superintegrable
Stäckel system. Then we investigate the Poisson algebras generated by the
Hamiltonians $h_{i}=E_{i}+W_{i}$ which we call _quasi-Stäckel Hamiltonians_.
Further, we show that the quasi-Stäckel Hamiltonians $h_{i}$ can be generated
from a new type of linear relations that generalize the separation relations
for the Stäckel Hamiltonians $E_{i}$. We call this new type of relations
_quasi-separation relations_ (which for $n=3$ are particular realizations of
the quasi-Stäckel systems from [17]). In Section 4 we prove a theorem (Theorem
4.1) which establishes an explicit relation between Poisson algebras of
$h_{i}$ and Lie-algebras of the corresponding non-homogeneous hydrodynamic
vector fields (1.1). In Section 5 we exploit the notion of Stäckel transform
[3, 4, 8, 14, 21] and analyze which of our systems $h_{i}$ can be mapped by
this transform to new systems $\tilde{h}_{i}$ in such a way that the
Hamiltonians $\tilde{h}_{i}$ also constitute an algebra. In this way we obtain
new non-homogeneous hydrodynamic equations.

Let us point out that this article does not deal with the problem of
integrability of the obtained non-homogeneous hydrodynamic equations. This is
a separate problem yet to investigate.

##  2 Homogeneous hydrodynamic Killing equations
generated by geodesic Stäckel systems

In this section we briefly remind some facts about the relation between
hydrodynamic Killing equations and geodesic Stäckel systems.

Consider the following $n$ separation relations

$\displaystyle\sum_{j=1}^{n}\Phi_{ij}(\lambda_{i})E_{j}=\frac{1}{2}f_{i}(\lambda_{i})\mu_{i}^{2},\qquad
i=1,\ldots,n,$ (2.1)

on $M=\mathbf{R}^{2n}$, where $\Phi_{ij}$ and $f_{i}$ are arbitrary functions
of one real variable. Since the $i$-th row of the matrix $\Phi(\lambda)$
depends on $\lambda_{i}$ only, it is the so called Stäckel matrix. The
variables
$(\lambda,\mu)=(\lambda_{1},\ldots,\lambda_{n},\mu_{1},\ldots,\mu_{n})$ will
be in the sequel referred to as position and momentum separation coordinates
on the phase space $M$. Solving the linear system (2.1) with respect to
$E_{j}$ yields $n$ quadratic in momenta functions (Hamiltonians) on $M$

$\displaystyle E_{r}=\frac{1}{2}\mu^{T}A_{r}\mu,\qquad r=1,\ldots,n,$ (2.2)

where $A_{r}$ are $n\times n$ matrices given by

$\displaystyle
A_{r}=\operatorname{diag}\big{(}f_{1}(\lambda_{1})\big{(}\Phi^{-1}\big{)}_{r1},\ldots,f_{n}(\lambda_{n})\big{(}\Phi^{-1}\big{)}_{rn}\big{)},\qquad
r=1,\ldots,n.$

By their construction the functions $E_{r}$ are in involution with respect to
the canonical Poisson bracket on $M$ given by
$\\{\lambda_{i},\mu_{j}\\}=\delta_{ij}$. They are referred to in literature as
_geodesic Stäckel Hamiltonians_. Further, we can factorize $A_{r}$ as
$A_{r}=K_{r}G$ with

$\displaystyle
G=A_{1}=\operatorname{diag}\big{(}f_{1}(\lambda_{1})\big{(}\Phi^{-1}\big{)}_{11},\ldots,f_{n}(\lambda_{n})\big{(}\Phi^{-1}\big{)}_{1n}\big{)}$
(2.3)

and with

$\displaystyle
K_{r}=\operatorname{diag}\left(\frac{\big{(}\Phi^{-1}\big{)}_{r1}}{\big{(}\Phi^{-1}\big{)}_{11}},\ldots,\frac{\big{(}\Phi^{-1}\big{)}_{rn}}{\big{(}\Phi^{-1}\big{)}_{1n}}\right),\qquad
r=1,\ldots,n$

(so that $K_{1}=I$). From now on we will interpret the matrix $G$ as a
contravariant form of a metric tensor on $M$. The corresponding covariant
metric tensor will be denoted by $g$ so that $gG=I$. It can be shown that the
matrices $K_{r}$ are then $(1,1)$-Killing tensors of the metric $G$. For a
fixed Stäckel matrix $\Phi$ we have thus the whole family of metrics $G$
parametrized by $n$ arbitrary functions $f_{i}$ of one variable. The tensors
$K_{r}$ are then Killing tensors for any metric from this family. Thus, the
Stäckel Hamiltonians $E_{r}$ are geodesic Hamiltonians of a Liouville
integrable system in the Riemannian space $(M,g)$. The Hamiltonian equations
for $E_{r}$ are given by

$\displaystyle\lambda_{t_{r}}=\frac{\partial
E_{r}}{\partial\mu},\qquad\mu_{t_{r}}=-\frac{\partial
E_{r}}{\partial\lambda},\qquad r=1,\dots,n,$ (2.4)

with the implicit common solution for all the positions
$\lambda_{i}=\lambda_{i}(t_{1},\dots,t_{n})$ in the form

$\displaystyle\sum_{k=1}^{n}\int^{\lambda_{k}}\frac{\Phi_{k,n-r}(\xi)}{\varphi_{k}(\xi)}d\xi=t_{r},\qquad
r=1,\dots,n,$ (2.5)

where

$\displaystyle\varphi_{k}(\xi)=\left(\frac{1}{2}f_{k}(\xi){\sum_{j=1}^{n}}\Phi_{kj}(\xi)a_{j}\right)^{\frac{1}{2}}$
(2.6)

and $a_{j}$ are arbitrary constants parametrizing the Liouville tori (values
of Hamiltonians $E_{r}$). The first part of Hamiltonian equations (2.4) can be
explicitly written as $\lambda_{t_{r}}=K_{r}G\mu$ so that
$G\mu=\lambda_{t_{1}}$ and every solution $\lambda_{i}(t_{1},\ldots,t_{n})$ of
(2.4) satisfies the following system of hydrodynamic equations

$\displaystyle\lambda_{t_{r}}=K_{r}(\lambda)\lambda_{x}\equiv
Z_{r}(\lambda,\lambda_{x}),\qquad r=2,\dots,n,$ (2.7)

where we denoted $t_{1}=x$. We call the system (2.7) a _hydrodynamic Killing
system_. The differential functions $Z_{r}$ can be interpreted as vector
fields on an infinite-dimensional space of functions $\lambda(x)$ and since
$\\{E_{i},E_{j}\\}=0$ one can show that also the vector fields $Z_{r}$
commute: $[Z_{r},Z_{s}]=0$ for all $r,s=1,\ldots,n$ (where $Z_{1}=\lambda_{x}$
is the translational symmetry). The system (2.7) is also known in the
literature as a weakly nonlinear semi-Hamilton system and since $K_{r}$ are
diagonal in $\lambda$-variables the variables $\lambda$ are in this context
known as _Riemann invariants_. The general solution of (2.7) is also given by
(2.5) if we allow $\varphi_{k}$ to be arbitrary. Therefore, due to (2.6), any
solution of the hydrodynamic system (2.7) can be obtained as a solution of a
particular Stäckel system with the suitably chosen functions $f_{i}$.

##  3 Quasi-Stäckel systems

The aim of this paper is to construct non-homogeneous hydrodynamic equations
that can still be related to some finite-dimensional integrable systems. The
first steps in that direction was made by Ferapontov and Fordy in [11, 13]. As
we wrote in Introduction we will modify the geodesic Hamiltonians $E_{i}$ by
linear in momenta constants of motion $W_{i}$ for Hamiltonian $E_{1}$, which
makes the corresponding hydrodynamic equations non-homogeneous, so that $(i)$
the modified system defines a maximally superintegrable system where the extra
integrals of motion are all linear in momenta and $(ii)$ it constitutes a non-
commutative Poisson algebra. As a consequence, the structure constants of this
algebra and the algebra of corresponding hydrodynamic vector fields will be
(up to a sign) the same.

It is well known that the linear in momenta function
$W=p^{T}Y=\sum_{i}p_{i}Y^{i}$ on $M$ is an integral of motion of $E_{1}$ if
and only if $Y=\sum_{i}Y^{i}(q)\frac{\partial}{\partial q_{i}}$ is a Killing
vector for the metric $G$. Therefore, we have to find enough Killing vectors
$Y_{r}$ of $G$ in order to perform our task. In analogy with the homogeneous
case, the first part of Hamiltonian equations (2.4) for the modified
Hamiltonians $h_{r}=E_{r}+W_{r}$ will then take the non-homogenous form

$\displaystyle\lambda_{t_{r}}=K_{r}(\lambda)\lambda_{x}+Y_{r}(\lambda)\equiv
Z_{r}(\lambda,\lambda_{x}),\qquad r=2,\dots,n,$ (3.1)

where $W_{r}=p^{T}Y_{r}$.

We will solely work with the metrics of constant curvature. It is well known
that the Lie algebra of Killing vectors of constant curvature metrics is of
the maximal dimension $n(n+1)/2$, but the problem of identifying all the
constant-curvature metrics of the form (2.3) is not solved yet, so we will
focus on some particular classes of metrics of the form (2.3) which are known
to be of constant curvature.

###  3.1 Benenti class of Stäckel systems

We will thus impose two restrictions on the metric $G$. Firstly, we will only
consider a class of Stäckel systems (2.1) given by the Stäckel matrix of the
very particular form $\Phi_{ij}=\lambda_{i}^{n-j}$. This results in the
following separation relations

$\displaystyle\sum\limits_{j=1}^{n}\lambda_{i}^{n-j}E_{j}=\frac{1}{2}f_{i}(\lambda_{i})\mu_{i}^{2},\qquad
i=1,\ldots,n.$ (3.2)

Such systems are called in literature _Benenti systems_. Moreover, in this
case the metric tensor (2.3) attains the explicit form

$\displaystyle
G=A_{1}=\operatorname{diag}\left(\frac{f_{1}(\lambda_{1})}{\Delta_{1}},\ldots,\frac{f_{n}(\lambda_{n})}{\Delta_{n}}\right),\qquad\Delta_{i}=\prod_{j\neq
i}^{n}\left(\lambda_{i}-\lambda_{j}\right),$

while the Killing tensors $K_{r}$ are

$\displaystyle
K_{r}=(-1)^{r+1}\operatorname*{diag}\left(\frac{\partial\sigma_{r}}{\partial\lambda_{1}},\dots,\frac{\partial\sigma_{r}}{\partial\lambda_{n}}\right),\qquad
r=1,\ldots,n,$ (3.3)

where $\sigma_{r}(\lambda)$ are elementary symmetric polynomials in $\lambda$.

Secondly, we will also assume that all $f_{i}$ are equal to the same monomial
of order not exceeding $n+1$

$\displaystyle f_{i}(\lambda_{i})=\lambda_{i}^{m},\qquad
m\in\\{0,\ldots,n+1\\},$

which renders $n+2$ metrics

$\displaystyle
G_{m}=\operatorname{diag}\left(\frac{\lambda_{1}^{m}}{\Delta_{1}},\ldots,\frac{\lambda_{n}^{m}}{\Delta_{n}}\right),\qquad
m\in\left\\{0,\ldots,n+1\right\\},$

with the corresponding geodesic Stäckel Hamiltonians

$\displaystyle E_{r}^{m}=\frac{1}{2}\mu^{T}K_{r}G_{m}\mu,\qquad r=1,\ldots,n.$
(3.4)

######  Remark 3.1.

It can be shown that the metric $G_{m}$ is flat for $m\in\\{0,\ldots,n\\}$ and
of constant curvature for $m=n+1$. The separation variables $\lambda_{i}$ can
in this case be considered as appropriate degenerations of Jacobi elliptic
coordinates, see [15].

In the separation coordinates (Riemann invariants) $\lambda$ the hydrodynamic
equations (2.7) take a particular form

$\displaystyle\frac{d\lambda}{dt_{r}}=(-1)^{r+1}\frac{\partial\sigma_{r}}{\partial\lambda}\lambda_{x},\qquad
r=1,\ldots,n.$ (3.5)

It turns out that the search for the linear in momenta integrals for $E_{1}$
is much easier in the coordinates

$\displaystyle q_{i}(\lambda)=(-1)^{i}\sigma_{i}(\lambda).$ (3.6)

We will refer to them as Viète coordinates. The corresponding conjugate
momenta $p(\lambda,\mu)$ on $M$ are given by

$\displaystyle
p_{i}=-\sum_{k=1}^{n}\frac{\lambda_{k}^{n-i}}{\Delta_{k}}\mu_{k},\qquad
i=1,\ldots,n.$ (3.7)

In $q$-variables the components of $G_{m}$ are polynomials in $q$ given by [5]

$\displaystyle\left(G_{0}\right)^{ij}=\sum_{k=0}^{n-1}q^{k}\delta_{n+k+1}^{i+j},$
$\displaystyle\left(G_{m}\right)^{ij}=\begin{cases}\displaystyle\sum\limits_{k=0}^{n-m-1}q_{k}\delta_{n-m+k+1}^{i+j},&i,j=1,\dots,n-m,\\\
\displaystyle-\sum\limits_{k=n-m+1}^{n}q_{k}\delta_{n-m+k+1}^{i+j},&i,j=n-m+1,\dots,n,\\\
0,&\text{otherwise},\end{cases}\qquad m=1,\dots,n,$ (3.8)
$\displaystyle(G_{n+1})^{ij}=q_{i}q_{j}-q_{i+j},\qquad i,j=1,\dots,n$

(where we denote $q_{0}=1$), while the hydrodynamic equations (2.7) attain the
form $q_{t_{r}}=K_{r}(q)q_{x}$ or, explicitly [2]

$\displaystyle\frac{dq_{j}}{dt_{r}}=(Z_{r})^{j}=(q_{j+r-1})_{x}+\sum_{k=1}^{j-1}\big{(}q_{k}(q_{j+r-k-1})_{x}-q_{j+r-k-1}(q_{k})_{x}\big{)},\qquad
r,j=1,\ldots,n.\\!\\!\\!\\!$ (3.9)

###  3.2 Killing vectors and additional integrals of motion

Let us define the following sets of indices:

$\displaystyle I_{1}^{m}=\\{2,\ldots,n-m+1\\},\qquad
I_{2}^{m}=\\{n-m+2,\ldots,n\\},\qquad m=0,\ldots,n+1.$

The search for linear in momenta extra integrals of motion for $E_{1}^{m}$
that are also linear in $q$ yields the following lemma.

######  Lemma 3.2.

For $m=0,\ldots,n-1$ the functions

$\displaystyle W_{r}^{m}=\sum_{i=1}^{r-1}iq_{r-i-1}p_{n-m-i+1},\qquad r\in
I_{1}^{m},$ (3.10)

and for $m=2,\ldots,n+1$ the functions

$\displaystyle W_{r}^{m}=\sum_{i=1}^{n-r+1}iq_{r+i-1}p_{n-m+i+1},\qquad r\in
I_{2}^{m}$ (3.11)

are constants of motion for $E_{1}^{m}$.

For each $m$ between $1$ and $n$, this lemma yields $n-1$ (and $n$ in case
$m=0$ and $m=n+1$) linear in both momenta $p$ and positions $q$ additional
constants of motion for $E_{1}^{m}$ with the corresponding Killing vector
fields $Y_{r}^{m}$ for $G_{m}$ so that

$\displaystyle
W_{r}^{m}=p^{T}Y_{r}^{m}=\sum_{i=1}^{n}p_{i}\left(Y_{r}^{m}\right)^{i}.$

######  Lemma 3.3.

For each $m\in\\{0,\ldots,n+1\\}$ the $2n-1$ functions

$\displaystyle E_{1}^{m},\ldots,E_{n}^{m},W_{2}^{m},\ldots,W_{n}^{m}$ (3.12)

constitute a maximally superintegrable system with respect to $E_{1}^{m}$.

###### Proof.

It is well known that if a Stäckel metric $G(q)$ on $Q\subset R^{n}$ is of
constant curvature then there exists $n-1$ independent second-order
contravariant Killing tensors $A_{2}(q),\dots,A_{n}(q)$ of metric $G$, which
commute with respect to the Schouten bracket (4.6): $[A_{i},A_{j}]_{S}=0,\
i,j=2,\dots,n$ and there exists $\frac{1}{2}n(n+1)$ Killing vectors $Y_{i}(q)$
of $G$. In consequence, on the phase space $T^{\ast}Q$, there exist $n$
functions quadratic in momenta
$E_{1}=\frac{1}{2}p^{T}G(q)p,E_{2}=\frac{1}{2}p^{T}A_{2}(q)p,\dots,E_{n}=\frac{1}{2}p^{T}A_{n}(q)p$
in involution with respect to the canonical Poisson bi-vector and
$\frac{1}{2}n(n+1)$ linear in momenta functions $W_{i}=p^{T}Y_{i}(q)$, forming
a Poisson algebra. In the set $(E,W)$ of $n+n(n+1)/2$ functions there exist
two types of functional relations: 1) all $E_{i}$ are expressible by
$W$-functions, i.e., $E_{i}=E_{i}(W)$, $i=1,\dots,n$ and 2 $\varphi_{k}(W)=0$,
where $\varphi_{k}$ is an appropriate number (depending on $n$) of functions,
quadratic in all $W$. Now, in order to prove the functional independence of
the subset (3.12), it is sufficient to show than none of these relations
survive in the set (3.12). The functional independence of $W_{2},\dots,W_{n}$
follows from the fact that the matrix of differentials
$dW_{2}^{m},\ldots,dW_{n}^{m}$ has for each $m$ the maximal rank $n-1$.
Further, since for each $m$ all $W_{i}^{m}$, $i=2,\dots,n$ contain together
less number of $q^{\prime}s$ then any $E_{j}$ does, so $E_{j}\neq
E_{j}(W_{2},\dots,W_{n})$, $j=1,\dots,n$. ∎

From (3.10), (3.11) and (3.6), (3.7) it follows that in the Riemann invariants
$\lambda$ the corresponding Killing vectors $Y_{r}^{m}$ of metrics $G_{m}$ are
represented by the formulas

$\displaystyle\big{(}Y_{r}^{m}\big{)}^{i}=\sum\limits_{k=1}^{r-1}(-1)^{r-k}k\sigma_{r-k-1}\frac{\lambda_{i}^{m+k-1}}{\Delta_{i}},\qquad
r\in I_{1}^{m}$ (3.13)

and

$\displaystyle\big{(}Y_{r}^{m}\big{)}^{i}=\sum\limits_{k=1}^{n-r+1}(-1)^{r+k}k\sigma_{r+k-1}\frac{\lambda_{i}^{m-k-1}}{\Delta_{i}},\qquad
r\in I_{2}^{m}.$ (3.14)

An important observation is that both $E_{r}^{m}$ and $W_{r}^{m}$ are
homogeneous functions of the same order $n-m-r+2$ with respect to the scaling

$\displaystyle p_{k}\rightarrow\varepsilon^{k}p_{k},\qquad
q_{k}\rightarrow\varepsilon^{-k}q_{k},$

so it is natural to consider their sums when constructing Hamiltonians with
linear in momenta terms. Let us thus define, for each $m$, the $n$ functions

$\displaystyle h_{1}^{m}=E_{1}^{m},\qquad h_{r}^{m}=E_{r}^{m}+W_{r}^{m},\qquad
r=2,\ldots,n.$ (3.15)

As we mentioned in introduction, we call the functions $h_{r}^{m}$ _quasi-
Stäckel Hamiltonians_.

######  Theorem 3.4.

The functions $h_{r}^{m}$, for each fixed $m$, constitute a Poisson algebra
$\mathfrak{g}=\operatorname{Span}\\{h_{r}^{m}\colon$ $r=1,\ldots,n\\}$ with
the following commutation relations for $i,j=2,\ldots,n$:

$\displaystyle\big{\\{}h_{i}^{m},h_{j}^{m}\big{\\}}=\begin{cases}0,&\text{for
}i\in I_{1}^{m}\text{ and }j\in I_{2}^{m},\\\
(j-i)h_{i+j-(n-m+2)}^{m},&\text{for }i,j\in I_{1}^{m},\\\
-(j-i)h_{i+j-(n-m+2)}^{m},&\text{for }i,j\in I_{2}^{m},\end{cases}$ (3.16)

where we use the notation $h_{i}^{m}=0$ for $i\leq 0$ or $i>n$.

Note that for the cases $n=2$ and $n=3$ this algebra is commutative.

###### Proof.

We have

$\displaystyle\big{\\{}h_{i}^{m},h_{j}^{m}\big{\\}}=\big{\\{}E_{i}^{m},E_{j}^{m}\big{\\}}+\big{\\{}E_{i}^{m},W_{j}^{m}\big{\\}}+\big{\\{}W_{i}^{m},E_{j}^{m}\big{\\}}+\big{\\{}W_{i}^{m},W_{j}^{m}\big{\\}},$
(3.17)

where the first term is zero as the Stäckel Hamiltonians $E_{i}^{m}$ pairwise
commute. By a direct calculation one can show that the functions $W_{i}^{m}$
themselves constitute a Poisson algebra with the same structure constants as
in (3.16). Finally, a calculation involving the formulas (3.8) and (3.9) shows
that for $i,j=2,\ldots,n$

$\displaystyle\big{\\{}E_{i}^{m},W_{j}^{m}\big{\\}}+\big{\\{}W_{i}^{m},E_{j}^{m}\big{\\}}=\begin{cases}0,&\text{for
}i\in I_{1}^{m}\text{ and }j\in I_{2}^{m},\\\
(j-i)E_{i+j-(n-m+2)}^{m},&\text{for }i,j\in I_{1}^{m},\\\
-(j-i)E_{i+j-(n-m+2)}^{m},&\text{for }i,j\in I_{2}^{m}.\end{cases}$ (3.18)

In consequence, (3.17) yields the right-hand side of (3.16). ∎

Due to the form of (3.16) the algebra $\mathfrak{g}$ splits into a direct sum
of two algebras $\mathfrak{g}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}$ where
$\mathfrak{g}_{1}=\operatorname{Span}\\{h_{i}\colon i\in I_{1}^{m}\\}$ and
$\mathfrak{g}_{2}=\operatorname{Span}\\{h_{i}\colon i\in I_{2}^{m}\\}$.

######  Remark 3.5.

The functions $h_{1}^{m},\ldots,h_{n}^{m},W_{2}^{m},\ldots,W_{n}^{m}$
constitute a non-commutative integrable system [7, 19].

###  3.3 Quasi-separation relations

Let us consider the following linear relations (cf. (2.1))

$\displaystyle{\displaystyle\sum\limits_{j=1}^{n}}\Phi_{ij}\left(\lambda_{i}\right)h_{j}=\frac{1}{2}f_{i}(\lambda_{i})\mu_{i}^{2}+\sum_{k=1}^{n}u_{ik}(\lambda)\mu_{k},\qquad
i=1,\ldots,n,$ (3.19)

with a given Stäckel matrix $\Phi$, with $n$ arbitrary functions $f_{i}$ of
one variable and with a set of $n^{2}$ functions $u_{ik}(\lambda)$ depending
in general on all $\lambda_{j}$. Similar relations were previously studied in
[17, 18]. Solving (3.19) with respect to $h_{i}$ we obtain

$\displaystyle h_{r}=E_{r}+W_{r},\qquad r=1,\ldots,n,$

such that $E_{r}$ are the geodesic Stäckel Hamiltonians (2.2), so they are in
mutual involution, and $W_{r}=p^{T}Y_{r}$ are some linear in momenta terms
(magnetic terms).

Assuming, additionally, that $(i)$ $W_{1}=0$ and $(ii)$ $\\{h_{1},h_{j}\\}=0$
for $j=2,\ldots,n$ we obtain that $Y_{r}$ are Killing vectors of the metric
$G$ in $E_{1}$. Thus, we obtain that $E_{1},\ldots,E_{n},W_{2},\ldots,W_{n}$
constitute a maximally superintegrable and separable system. In such a case
the relations (3.19) will be called _quasi-separation relations_.

######  Theorem 3.6.

The maximally superintegrable system (3.12) is generated by the following
quasi-separation relations:

$\displaystyle\sum_{j=1}^{n}\lambda_{i}^{n-j}h_{j}=\frac{1}{2}\lambda_{i}^{m}\mu_{i}^{2}+\sum_{k=1}^{n}u_{ik}(\lambda)\mu_{k},\qquad
i=1,\ldots,n,$ (3.20)

where

$\displaystyle\sum_{k=1}^{n}u_{ik}(\lambda)\mu_{k}=\begin{cases}\displaystyle-\sum_{k\neq
i}\frac{\mu_{i}-\mu_{k}}{\lambda_{i}-\lambda_{k}},&\text{for }m=0,\\\
\displaystyle-\lambda_{i}^{m-1}\sum_{k\neq
i}\frac{\lambda_{i}\mu_{i}-\lambda_{k}\mu_{k}}{\lambda_{i}-\lambda_{k}}+(m-1)\lambda_{i}^{m-1}\mu_{i},&\text{for
}m=1,\ldots,n,\\\ \displaystyle-\lambda_{i}^{n-1}\sum_{k\neq
i}\frac{\lambda_{i}^{2}\mu_{i}-\lambda_{k}^{2}\mu_{k}}{\lambda_{i}-\lambda_{k}}+(n-1)\lambda_{i}^{n}\mu_{i},&\text{for
}m=n+1.\end{cases}$ (3.21)

Thus, by solving (3.20), (3.21) with respect to $h_{j}$ we obtain all the
Hamiltonians (3.4) as well as the additional constants $W_{i}^{m}$ (3.10),
(3.11) of the Hamiltonian $E_{1}^{m}$.

##  4 Non-homogeneous hydrodynamic equations of Killing type

In this section we prove a theorem describing the relation between the systems
presented in Section 3 and non-homogeneous hydrodynamic equations. Consider
the set of Hamiltonians on $\mathbf{R}^{2n}$

$\displaystyle h_{r}=E_{r}+W_{r},\qquad E_{r}=\tfrac{1}{2}p^{T}A_{r}p,\qquad
W_{r}=p^{T}Y_{r},\qquad r=1,\dots,n,$ (4.1)

with

$\displaystyle\\{E_{r},E_{s}\\}=0,\qquad r,s=1,\ldots,n,$

where $A_{r}=K_{r}G$ with $L_{Y_{r}}G=0$ (i.e., all $Y_{r}$ are Killing
vectors for the metric $G$). Suppose also that the functions $W_{r}$
constitute a non-commutative Poisson algebra with some structure constants
$c_{rs}^{i}$

$\displaystyle\\{W_{r},W_{s}\\}=\sum_{i=1}^{n}c_{rs}^{i}W_{i},$ (4.2)

and moreover that the following condition holds

$\displaystyle\\{E_{r},W_{s}\\}+\\{W_{r},E_{s}\\}=\sum_{i=1}^{n}c_{rs}^{i}E_{i}.$
(4.3)

The conditions (4.2) and (4.3) imply that $h_{r}$ also constitute a non-
commutative Poisson algebra with the same structure constants $c_{rs}^{i}$

$\displaystyle\\{h_{r},h_{s}\\}=\sum_{i=1}^{n}c_{rs}^{i}h_{i}.$ (4.4)

In order to relate the algebra (4.4) with an appropriate algebra of
hydrodynamic vector fields we will use the link between the canonical Poisson
bracket and the Schouten bracket between symmetric contravariant tensors.
Actually, for a pair of functions on $M$

$\displaystyle F_{K}=\frac{1}{k!}K^{i_{1}\dots i_{k}}(q)p_{i_{1}}\cdots
p_{i_{k}},\qquad F_{R}=\frac{1}{r!}R^{i_{1}\dots i_{r}}(q)p_{i_{1}}\cdots
p_{i_{r}}$

(we use for the moment the Einstein summation convention) the following
relation holds [9]

$\displaystyle\\{F_{K},F_{R}\\}=-([K,R]_{S})^{i_{1}\dots
i_{k+r-1}}p_{i_{1}}\cdots p_{i_{k+r-1}},$ (4.5)

where

$\displaystyle([K,R]_{S})^{l_{1}\dots
l_{k+r-1}}=\frac{1}{k!r!}\big{[}kK^{i(l_{1}\dots}\partial_{n}R^{\dots
l_{k+r-1})}-rR^{i(l_{1}\dots}\partial_{i}K^{\dots
l_{k+r-1})}\big{]},\qquad\partial_{i}=\frac{\partial}{\partial q_{i}}$ (4.6)

defines the Schouten bracket on $\mathbf{R}^{n}$ (the configuration space) and
where $(\dots)$ is the symmetrization operation over the indices.

Formula (4.5) implies that the vector fields $Y_{r}$ constitute a non-abelian
Lie algebra with the structure constants $-c_{rs}^{i}$

$\displaystyle[Y_{r},Y_{s}]=-\sum_{i=1}^{n}c_{rs}^{i}Y_{i}$ (4.7)

and that

$\displaystyle[A_{r},Y_{s}]_{S}+[Y_{r},A_{s}]_{S}=-\sum_{i=1}^{n}c_{rs}^{i}A_{i}.$
(4.8)

From the properties of the Schouten bracket it follows that
$[Y_{r},A_{s}]_{S}=L_{Y_{r}}A_{s}$. Since moreover
$L_{Y_{s}}A_{r}=(L_{Y_{s}}K_{r})G$, as $Y_{s}$ are Killing vectors of a non-
singular metric $G$, we obtain from (4.8)

$\displaystyle L_{Y_{r}}K_{s}-L_{Y_{s}}K_{r}=-\sum_{i=1}^{n}c_{rs}^{i}K_{i}.$
(4.9)

######  Theorem 4.1.

Consider the non-homogeneous hydrodynamic systems

$\displaystyle q_{t_{r}}=K_{r}q_{x}+Y_{r}\equiv Z_{r}\qquad r=2,\ldots,n,$
(4.10)

where $K_{r}$ and $Y_{r}$ are such that the conditions (4.1)–(4.3) are
satisfied. Then

$\displaystyle[Z_{r},Z_{s}]=-\sum_{i=1}^{n}c_{rs}^{i}Z_{i}.$

###### Proof.

We have

$\displaystyle[Z_{r},Z_{s}]=[K_{r}q_{x}+Y_{r},K_{s}q_{x}+Y_{s}]=[K_{r}q_{x},K_{s}q_{x}]+[K_{r}q_{x},Y_{s}]+[Y_{r},K_{s}q_{x}]+[Y_{r},Y_{s}].$

To begin with, $\left[K_{r}q_{x},K_{s}q_{x}\right]=0$ for all $r$, $s$ (see
for example [12]). Further, as $[Y_{s},q_{x}]=0$ we have
$[Y_{s},K_{r}q_{x}]=(L_{Y_{s}}K_{r})q_{x}$. Using (4.7) and (4.9) we obtain

$\displaystyle[Z_{r},Z_{s}]=(L_{Y_{r}}K_{s}-L_{Y_{s}}K_{r})q_{x}-\sum_{i=1}^{n}c_{rs}^{i}Y_{i}=-\sum_{i=1}^{n}c_{rs}^{i}(K_{i}q_{x}+Y_{i})=-\sum_{i=1}^{n}c_{rs}^{i}Z_{i}.$
∎

Thus, the vector fields $Z_{r}$ in (4.10) constitute a Lie algebra with up to
a sign the same structure constants as the Poisson algebra (4.4). Besides,
since our systems (3.15) with structure constants (3.16) satisfy the
conditions (4.1)–(4.3) (as (3.18) is a specification of (4.3)) we see that
they possess non-homogeneous hydrodynamic counterparts, with the same
structure constants.

######  Example 4.2.

Let us consider the case $n=4$ in Viète coordinates $q$. As $m=0,\ldots,n+1$
we have then $n+2=6$ different non-homogeneous hydrodynamic systems (4.10):

$\displaystyle\left[\begin{matrix}q_{1}\\\ q_{2}\\\ q_{3}\\\
q_{4}\end{matrix}\right]_{t_{2}}=\left[\begin{matrix}0&1&0&0\\\
-q_{2}&q_{1}&1&0\\\ -q_{3}&0&q_{1}&1\\\
-q_{4}&0&0&q_{1}\end{matrix}\right]\left[\begin{matrix}q_{1}\\\ q_{2}\\\
q_{3}\\\ q_{4}\end{matrix}\right]_{x}+Y_{2}\equiv Z_{2},$
$\displaystyle\left[\begin{matrix}q_{1}\\\ q_{2}\\\ q_{3}\\\
q_{4}\end{matrix}\right]_{t_{3}}=\left[\begin{matrix}0&0&1&0\\\
-q_{3}&0&q_{1}&1\\\ -q_{4}&-q_{3}&q_{2}&q_{1}\\\
-0&-q_{4}&0&q_{2}\end{matrix}\right]\left[\begin{matrix}q_{1}\\\ q_{2}\\\
q_{3}\\\ q_{4}\end{matrix}\right]_{x}+Y_{3}\equiv Z_{3},$
$\displaystyle\left[\begin{matrix}q_{1}\\\ q_{2}\\\ q_{3}\\\
q_{4}\end{matrix}\right]_{t_{4}}=\left[\begin{matrix}0&0&0&1\\\
-q_{4}&0&0&q_{1}\\\ 0&-q_{4}&0&q_{2}\\\
0&0&-q_{4}&q_{3}\end{matrix}\right]\left[\begin{matrix}q_{1}\\\ q_{2}\\\
q_{3}\\\ q_{4}\end{matrix}\right]_{x}+Y_{4}\equiv Z_{4},$

where

$\displaystyle\text{for }m=0\colon\quad$ $\displaystyle
Y_{2}=\left(0,0,0,1\right)^{T},\qquad Y_{3}=(0,0,2,q_{1})^{T},\qquad
Y_{4}=(0,3,2q_{1},q_{2})^{T},$ $\displaystyle\text{for }m=1\colon\quad$
$\displaystyle Y_{2}=(0,0,1,0)^{T},\qquad Y_{3}=(0,2,q_{1},0)^{T},\qquad
Y_{4}=(3,2q_{1},q_{2},0)^{T},$ $\displaystyle\text{for }m=2\colon\quad$
$\displaystyle Y_{2}=(0,1,0,0)^{T},\qquad Y_{3}=(2,q_{1},0,0)^{T},\qquad
Y_{4}=(0,0,0,q_{4})^{T},$ $\displaystyle\text{for }m=3\colon\quad$
$\displaystyle Y_{2}=(1,0,0,0)^{T},\qquad Y_{3}=(0,0,q_{3},2q_{4}),\qquad
Y_{4}=(0,0,q_{4},0)^{T},$ $\displaystyle\text{for }m=4\colon\quad$
$\displaystyle Y_{2}=(0,q_{2},2q_{3},3q_{4})^{T},\qquad
Y_{3}=(0,q_{3},2q_{4},0)^{T},\qquad Y_{4}=(0,q_{4},0,0)^{T},$
$\displaystyle\text{for }m=5\colon\quad$ $\displaystyle
Y_{2}=(q_{2},2q_{3},3q_{4},0)^{T},\qquad Y_{3}=(q_{3},2q_{4},0,0)^{T},\qquad
Y_{4}=(q_{4},0,0,0)^{T},$

which constitute appropriate Lie algebras with the following nonzero elements
given by Theorem 4.1 and (3.16)

$\displaystyle\text{for }m=0\colon\quad$ $\displaystyle[Z_{3},Z_{4}]=-Z_{1},$
$\displaystyle\text{for }m=1\colon\quad$
$\displaystyle[Z_{2},Z_{4}]=-2Z_{1},\qquad[Z_{3},Z_{4}]=-Z_{2},$
$\displaystyle\text{for }m=2\colon\quad$ $\displaystyle[Z_{2},Z_{3}]=-Z_{1},$
$\displaystyle\text{for }m=3\colon\quad$ $\displaystyle[Z_{3},Z_{4}]=Z_{4},$
$\displaystyle\text{for }m=4\colon\quad$
$\displaystyle[Z_{2},Z_{3}]=Z_{3},\qquad[Z_{2},Z_{4}]=2Z_{4},$
$\displaystyle\text{for }m=5\colon\quad$ $\displaystyle[Z_{2},Z_{3}]=Z_{3},$

where $Z_{1}=(q_{1},q_{2},q_{3},q_{4})_{x}^{T}$. Using (3.5), (3.13) and
(3.14) the present example can be easily calculated in the Riemman invariants
$\lambda$.

##  5 Stäckel transform and new non-homogeneous hydrodynamic Killing systems

Stäckel transform is a functional transform that maps a Liouville integrable
system into a new integrable system, and in particular it maps a Stäckel
system into a new Stäckel systems [3, 8, 14, 21], which explains its name. In
[4] the authors considered the action of Stäckel transform on superintegrable
systems in such a way that it preserves superintegrability. It was found that
only particular one-parameter Stäckel transforms preserve superintegrability.
Here we demonstrate that Stäckel transforms are also applicable for our
particular systems (3.15) defined by the quasi-separation relations (3.20).
Nevertheless, if we demand that the transformed system shall also constitute a
Poisson algebra, then the number of admissible Stäckel transforms becomes very
limited.

Consider thus Hamiltonians (3.15) extended by some potentials $V_{r}^{(k)}$

$\displaystyle h_{1}^{m}=E_{1}^{m}+\alpha V_{1}^{(k)}\equiv H_{1}^{m},$
$\displaystyle h_{r}^{m}=E_{r}^{m}+\alpha V_{r}^{(k)}+W_{r}^{m}\equiv
H_{r}^{m}+W_{r}^{m},\qquad r=2,\ldots,n,$ (5.1)

where $\alpha$ is a parameter, and where the potentials $V_{r}^{(k)}$,
$k\in\mathbf{Z}$, are defined by the following separation relations

$\displaystyle\lambda_{i}^{k}+\sum_{j=1}^{n}\lambda_{i}^{n-j}V_{j}^{(k)}=0,\qquad
i=1,\ldots,n.$ (5.2)

As such, the potentials $V_{r}^{(k)}$ are called basic separable potentials.
Thus, the functions $h_{r}^{m}$ in (5.1) are generated by the following quasi-
separation relations

$\displaystyle\alpha\lambda_{i}^{k}+\sum_{j=1}^{n}\lambda_{i}^{n-j}h_{j}=\frac{1}{2}\lambda_{i}^{m}\mu_{i}^{2}+\sum_{k=1}^{n}u_{ik}(\lambda)\mu_{k},\qquad
i=1,\ldots,n,$ (5.3)

with $u_{ik}$ as in (3.21), i.e., the term $\lambda_{i}^{k}$ in (5.3)
generates the potential $V_{r}^{(k)}$ in (5.1).

######  Lemma 5.1.

The functions $H_{1}^{m},\ldots,H_{n}^{m},W_{2}^{m},\ldots,W_{n}^{m}$
constitute a maximally superintegrable $($with respect to $H_{1}^{m})$ system
only in the following four cases: $(m,k)=(0,n),(n,n),(1,-1)$ and $(n+1,-1)$.

###### Proof.

Since the functions $H_{r}^{m}$ are Stäckel Hamiltonians (albeit no longer
geodesic), they commute: $\\{H_{r}^{m},H_{s}^{m}\\}=0$ for any $r$, $s$, so
$H_{1}^{m},\ldots,H_{n}^{m}$ constitute a Liouville integrable system.
Moreover, $\\{E_{1}^{m},W_{r}^{m}\\}=0$ for any $r$ due to Lemma 3.2. Thus, in
order to have $\\{H_{1}^{m},W_{r}^{m}\\}=0$ for any $r$ we have to demand
$\\{V_{1}^{(k)},W_{r}^{m}\\}=0$ or, equivalently,
$L_{Y_{r}^{m}}(V_{1}^{(k)})=0$, for all $r$ which is satisfied only in the
mentioned four cases. ∎

Let us now turn to algebraic properties of the set of functions $h_{r}^{m}$ in
(5.1). Since some of these functions can now commute to the constant $\alpha$
we have to extend the set of function $h_{r}^{m}$ by the constant Hamiltonian
$h_{0}=\alpha$ in order to turn it into a Poisson algebra.

######  Proposition 5.2.

The functions $h_{0}^{m}=\alpha,h_{1}^{m},\ldots,h_{n}^{m}$ in (5.1) for
$(m,k)=(0,n),(n,n),(1,-1)$ and $(n+1,-1)$ constitute a Poisson algebra with
the commutation relations as in (3.16).

Let us note that since the potentials $V_{r}^{(k)}$ in (5.1) do not appear in
the first part of Hamiltonian equations (2.4) they do not appear in the
corresponding hydrodynamic system (4.10). Nor does the constant Hamiltonian
$h_{0}^{m}=\alpha$. Therefore, the hydrodynamic system corresponding to (3.15)
and the one corresponding to (5.1) are exactly the same.

Due to Lemma 5.1, we will on what follows only need the potentials
$V_{r}^{(n)}$ and $V_{r}^{(-1)}$. They can be calculated from (5.2) and are in
$q$-coordinates given by

$\displaystyle V_{r}^{(n)}=q_{r},\qquad
V_{r}^{(-1)}=\frac{q_{r-1}}{q_{n}},\qquad r=1,\ldots,n.$

Let us now perform the Stäckel transform of the Hamiltonians $h_{r}^{m}$ in
(5.1) with respect to the parameter $\alpha$. It means that we first solve the
relation $h_{1}^{m}=\tilde{\alpha}$, i.e., $E_{1}^{m}+\alpha
V_{1}^{(k)}=\tilde{\alpha}$, with respect to $\alpha$ which yields

$\displaystyle\tilde{h}_{1}^{m}=\alpha=-\frac{1}{V_{1}^{(k)}}E_{1}^{m}+\tilde{\alpha}\frac{1}{V_{1}^{(k)}}$

and then replace $\alpha$ with $\tilde{h}_{1}^{m}$ in all the remaining
Hamiltonians $h_{r}^{m}$, $r=2,\ldots,n$, which yields

$\displaystyle\tilde{h}_{r}^{m}=h_{r}^{m}\big{|}_{\alpha\rightarrow\tilde{h}_{1}^{m}}=E_{r}^{m}+\left(-\frac{1}{V_{1}^{(k)}}E_{1}^{m}+\tilde{\alpha}\frac{1}{V_{1}^{(k)}}\right)V_{r}^{(k)}+W_{r}^{m}.$

Thus, the Stäckel transform of (5.1) with respect to $\alpha$ attains the form

$\displaystyle\tilde{h}_{1}^{m}=-\frac{1}{V_{1}^{(k)}}E_{1}^{m}+\tilde{\alpha}\frac{1}{V_{1}^{(k)}},$
(5.4)
$\displaystyle\tilde{h}_{r}^{m}=E_{r}^{m}-\frac{V_{r}^{(k)}}{V_{1}^{(k)}}E_{1}^{m}+\tilde{\alpha}\frac{V_{r}^{(k)}}{V_{1}^{(k)}}+W_{r}^{m}=\tilde{H}_{r}^{m}+W_{r}^{m},\qquad
r=2,\dots,n,$ (5.5)

where

$\displaystyle\tilde{H}_{r}^{m}=\tilde{E}_{r}^{m}+\tilde{\alpha}\frac{V_{r}^{(k)}}{V_{1}^{(k)}},\qquad\tilde{E}_{r}^{m}=\frac{1}{2}p^{T}\tilde{K}_{r}\tilde{G}_{m}p.$

Due to (5.4) the metric $\tilde{G}_{m}$ in $\tilde{h}_{1}^{m}$ is of the form

$\displaystyle\tilde{G}_{m}=-\frac{1}{V_{1}^{(k)}}G_{m}.$

Since $\tilde{G}_{m}$ is a conformal deformation of the metric $G_{m}$ by
$V_{1}^{(k)}$, which in the considered four cases
$(m,k)=(0,n),(n,n),(1,-1),(n+1,-1)$ satisfies $L_{Y_{r}^{m}}(V_{1}^{(k)})=0$,
all $n-1$ vector fields $Y_{r}^{m}$ are in these four cases Killing vectors
for the metric $\tilde{G}_{m}$ as well. Thus, the corresponding functions
$W_{r}^{m}=p^{T}Y_{r}^{m}$ are constants of motion not only for $H_{1}^{m}$
but also for $\tilde{H}_{1}^{m}$ and therefore the functions
$\tilde{H}_{1}^{m},\ldots,\tilde{H}_{n}^{m},W_{2}^{m},\ldots,W_{n}^{m}$ also
constitute a maximally superintegrable system.

Due to the definition of our Stäckel transform (5.5), on the level of
separation relations (5.3) the Stäckel transform renders the following
substitution:

$\displaystyle\alpha\rightarrow\tilde{h}_{1}^{m},\qquad
h_{1}\rightarrow\tilde{\alpha},\qquad
h_{r}^{m}\rightarrow\tilde{h}_{r}^{m}\qquad\text{for}\quad r=2,\ldots,n.$
(5.6)

In consequence, the quasi-separation relations for our four cases have the
form

$\displaystyle\tilde{\alpha}\lambda_{i}^{n-1}+\lambda_{i}^{k}\tilde{h}_{1}^{m}+\sum_{j=2}^{n}\lambda_{i}^{n-j}\tilde{h}_{j}^{m}=\frac{1}{2}\lambda_{i}^{m}\mu_{i}^{2}+\sum_{s=1}^{n}u_{is}(\lambda)\mu_{s},\qquad
i=1,\ldots,n,$ (5.7)

with $u_{ik}$ as in (3.21) (compare with (5.3)). However, applying the
canonical transformation $\lambda_{i}\rightarrow\lambda_{i}^{-1}$,
$\mu_{i}\rightarrow-\lambda_{i}^{2}\mu_{i}$ to the quasi-separation relations
(5.7) with $(m,k)=(1,-1)$ one obtains the quasi-separation relations (3.20)
with $m=n+1$ while in the case $(m,k)=(n+1,-1)$ we receive after this
transformation the quasi-separation relations (3.20) with $m=1$ so they can
not be considered as new systems. Let us thus focus on two remaining cases:
$(m,k)=(0,n)$ and $(n,n)$. For both these cases the Killing tensors
$\tilde{K}_{r}$ can be expressed through the Killing tensors $K_{r}$ (3.3) as
[1]

$\displaystyle\tilde{K}_{r}=K_{r+1}-K_{2}K_{r},\qquad r=1,\ldots,n.$ (5.8)

Let us now find whether the functions $\tilde{h}_{r}^{m}$ also constitute a
Poisson algebra. As it has been demonstrated in [3]

$\displaystyle\\{\tilde{h}_{r},\tilde{h}_{s}\\}=\sum_{i,j=1}^{n}\big{(}A^{-1}\big{)}_{ri}\big{(}A^{-1}\big{)}_{sj}\\{h_{i},h_{j}\\},$
(5.9)

where the $n\times n$ matrix $A$ is of the form

$\displaystyle A_{ij}=\delta_{ij},\qquad j=2,\ldots,n,\qquad
A_{i1}=-\frac{\partial h_{i}}{\partial\alpha}=-V_{i}^{(k)},\qquad
i=1,\ldots,n.$

So, in general, there is no guarantee that the functions $\tilde{h}_{r}$
constitute an algebra even if $h_{r}$ do.

######  Proposition 5.3.

In the two cases $(m,k)=(0,n)$ and $(n,n)$ the functions $\tilde{h}_{r}^{m}$
constitute a Poisson algebra, with the structure constants obtained from the
structure constants of the algebra of $h_{r}^{m}$ given in Proposition 5.2
through the substitution (5.6).

To see this, it is enough to realize that in both cases $(k,m)=(0,n)$ and
$(n,n)$ the formula (5.9) reduces to

$\displaystyle\big{\\{}\tilde{h}_{r}^{m},\tilde{h}_{s}^{m}\big{\\}}=\big{\\{}h_{r}^{m},h_{s}^{m}\big{\\}}\qquad\text{for
all}\ \ r,\,s,$ (5.10)

while the substitution (5.6) in the right-hand side of (5.10) allows for
expressing $\\{h_{r}^{m},h_{s}^{m}\\}$ in terms of the Hamiltonians
$\tilde{h}_{r}^{m}$.

######  Example 5.4.

For $n=5$, $m=0$ and $k=n$ the Hamiltonians $h_{r}^{m}$ in (5.1) constitute a
Poisson algebra with the following non-zero brackets:

$\displaystyle\\{h_{2},h_{5}\\}=3\alpha,\qquad\\{h_{3},h_{4}\\}=\alpha,\qquad\\{h_{3},h_{5}\\}=2h_{1},\qquad\\{h_{4},h_{5}\\}=h_{2},$

and after the Stäckel transform (5.5) the corresponding algebra of
$\tilde{h}_{1}^{m}$ has due to (5.6) the following non-zero brackets

$\displaystyle\big{\\{}\tilde{h}_{2},\tilde{h}_{5}\big{\\}}=3\tilde{h}_{1},\qquad\big{\\{}\tilde{h}_{3},\tilde{h}_{4}\big{\\}}=\tilde{h}_{1},\qquad\big{\\{}\tilde{h}_{3},\tilde{h}_{5}\big{\\}}=2\tilde{\alpha},\qquad\big{\\{}\tilde{h}_{4},\tilde{h}_{5}\big{\\}}=\tilde{h}_{2}.$

Our two cases, that is $(m,k)=(0,n)$ or $(n,n)$, lead after the Stäkel
transform to two new non-homogeneous hydrodynamic systems through the Theorem
4.1. They have the form

$\displaystyle q_{t_{r}}=\tilde{K}_{r}q_{x}+Y_{r}\equiv\tilde{Z}_{r},\qquad
r=2,\ldots,n,$ (5.11)

with $Y_{r}$ for $m=0$ and $m=n$ the same as before and with $\tilde{K}_{r}$
given by (5.8).

######  Example 5.5.

For $n=4$ the systems (5.11) attain in Viète coordinates $q$ the form

$\displaystyle\left[\begin{matrix}q_{1}\\\ q_{2}\\\ q_{3}\\\
q_{4}\end{matrix}\right]_{t_{2}}=\left[\begin{matrix}q_{2}&-q_{1}&0&0\\\
q_{1}q_{2}&q_{2}-q_{1}^{2}&-q_{1}&0\\\ q_{1}q_{3}&0&q_{2}-q_{1}^{2}&-q_{1}\\\
q_{1}q_{4}&0&0&q_{2}-q_{1}^{2}\end{matrix}\right]\left[\begin{matrix}q_{1}\\\
q_{2}\\\ q_{3}\\\ q_{4}\end{matrix}\right]_{x}+Y_{2}=\tilde{Z}_{2},$
$\displaystyle\left[\begin{matrix}q_{1}\\\ q_{2}\\\ q_{3}\\\
q_{4}\end{matrix}\right]_{t_{3}}=\left[\begin{matrix}q_{3}&0&-q_{1}&0\\\
q_{1}q_{3}&q_{3}&-q_{1}^{2}&-q_{1}\\\
q_{1}q_{4}&q_{1}q_{3}&q_{3}-q_{1}q_{2}&-q_{1}^{2}\\\
0&q_{1}q_{4}&0&q_{3}-q_{1}q_{2}\end{matrix}\right]\left[\begin{matrix}q_{1}\\\
q_{2}\\\ q_{3}\\\ q_{4}\end{matrix}\right]_{x}+Y_{3}=\tilde{Z}_{3},$
$\displaystyle\left[\begin{matrix}q_{1}\\\ q_{2}\\\ q_{3}\\\
q_{4}\end{matrix}\right]_{t_{4}}=\left[\begin{matrix}q_{4}&0&0&-q_{1}\\\
q_{1}q_{4}&q_{4}&0&-q_{1}^{2}\\\ 0&q_{1}q_{4}&q_{4}&-q_{1}q_{2}\\\
0&0&q_{1}q_{4}&q_{4}-q_{1}q_{3}\end{matrix}\right]\left[\begin{matrix}q_{1}\\\
q_{2}\\\ q_{3}\\\ q_{4}\end{matrix}\right]_{x}+Y_{4}=\tilde{Z}_{4},$

where

$\displaystyle\text{for }m=0\colon\quad$ $\displaystyle
Y_{2}=(0,0,0,1)^{T},\qquad Y_{3}=(0,0,2,q_{1})^{T},\qquad
Y_{4}=(0,3,2q_{1},q_{2})^{T},$ $\displaystyle\text{for }m=4\colon\quad$
$\displaystyle Y_{2}=(0,q_{2},2q_{3},3q_{4})^{T},\qquad
Y_{3}=(0,q_{3},2q_{4},0)^{T},\qquad Y_{4}=(0,q_{4},0,0)^{T},$

which constitute Lie algebras with the following nonzero elements given by
Theorem 4.1 and (3.16)

$\displaystyle m=0\colon\quad$
$\displaystyle[\tilde{Z}_{2},\tilde{Z}_{4}]=-2\tilde{Z}_{1},$ $\displaystyle
m=4\colon\quad$
$\displaystyle[\tilde{Z}_{2},\tilde{Z}_{3}]=\tilde{Z}_{3},\qquad[\tilde{Z}_{2},\tilde{Z}_{4}]=2\tilde{Z}_{4},$

where $\tilde{Z}_{1}=(q_{1},q_{2},q_{3},q_{4})_{x}^{T}$.

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