The vacuum and electrovacuum Einstein equations for spacetimes with two commuting Killing vectors can be solved by indirect methods of integrable systems. But if in addition the spacetime admits an irreducible Killing tensor and the corresponding KleinGordon equation is separable they can be integrated directly by separation of variables as shown by Carter in 1968. We generalize this approach to supergravity and derive a metric ansatz that ensures the above properties for Petrovtype I. Our derivation is based on the BenentiFrancavilla ansatz for metrics admitting two commuting Killing vectors and an irreducible Killing tensor. We find additional constraints that guarantee the existence of two shearfree null geodesic congruences and the separability of the KleinGordon equation. The resulting class of metrics belongs to a certain sector of Petrov type I called IB whose algebraically special subsector contains only type D. For this class a direct integration of the supergravity equations seems possible. We also show that these spacetimes admit a general description of the generalized photon and massive particle surfaces recently introduced in connection with black hole shadows.
['General Relativity and Quantum Cosmology', 'High Energy Physics - Theory', 'Mathematical Physics']
Petrov types separability and generalized photon surfaces of supergravity black holes
{'Petrov types, separability and generalized photon surfaces of supergravity black holes': "Dmitri Gal'tsov ∗ and Aleksandr Kulitskii †  \nFaculty of Physics, Moscow State University, 119899, Moscow, Russia", 'Abstract': 'The vacuum and electrovacuum Einstein equations for spacetimes with two commuting Killing vectors can be solved by indirect methods of integrable systems. But if, in addition, the spacetime admits an irreducible Killing tensor and the corresponding Klein-Gordon equation is separable, they can be integrated directly by separation of variables, as shown by Carter in 1968. We generalize this approach to supergravity and derive a metric ansatz that ensures the above properties for Petrov-type  \n- I . Our derivation is based on the Benenti-Francavilla ansatz for metrics admitting two commuting Killing vectors and an irreducible Killing tensor. We find additional constraints that guarantee the  \nexistence of two shear-free null geodesic congruences and the separability of the Klein-Gordon equation.  \nThe resulting class of metrics belongs to a certain sector of Petrov type I , called I B , whose algebraically special subsector contains only type D . For this class, a direct integration of the supergravity equations seems possible. We also show that these spacetimes admit a general description of the generalized photon and massive particle surfaces recently introduced in connection with black hole shadows.', 'I. INTRODUCTION': 'Hidden symmetries of spacetime, exhibited by second-rank Killing tensors, play no less an important role than isometries; for a review, see [1-3]. Over the past decade, significant progress has been made in the analysis of geometries admitting Killing-Yano (KY) tensors [2-4], which are the strongest symmetries in the Killing hierarchy for D -type spacetimes.  \nHidden symmetries of spacetimes beyond type D , which do not allow for KY structures, have also been examined, but their analysis is not yet so complete. Here we would like to investigate hidden symmetries of black holes in supergravities N = 4 , 8, which include scalar fields and belong to general Petrov type I . The exact solutions for stationary black holes in these theories are fairly well known, largely due to hidden symmetries of a different kind that arise when these theories are dimensionally reduced in stationary spacetimes. As is well known, the dimensional reduction of supergravities to three dimensions leads to sigma models on homogeneous target spaces [5-7]. The target space isometries include Harrison transformations, which allow generating charged supergravity black holes from the Kerr metric. Apart from static solutions, which are easily obtained directly from the Einstein equations, most of the known stationary solutions were found just in this way, see e.g. [8-11]. A few exceptions are the supersymmetric solutions obtained using the Bogomolny equations [12, 13] or solving the null geodesic equations in the target space [14] (also known as the nilpotent orbit method [15]).  \nMeanwhile, in vacuum and electrovacuum gravity, Carter showed that direct integration of stationary axisymmetric Einstein equations is possible for spacetimes admitting both HamiltonJacobi and Klein-Gordon separability [16]. By explicit integration he found several classes of solutions, among which were Kerr and Kerr-Newman black holes of the Petrov type D (for some details of the electromagnetic dressing see also [17]).  \nBlack holes in four-dimensional extended supergravities N = 4 , 8 generically belong to Petrov type I , though they still admit the second rank Killing tensor like their type D cousins in vacuum and electrovacuum gravity. We use the Benenti-Francaviglia (BF) [18] parameterization of metrics admitting a Killing tensor independently of their Petrov type (Section 2). We then require for them the fulfillment of another important property of stationary axisymmetric spacetimes which holds in type D : the existence of two null geodesic shearfree congruences (Section 3). This is accomplished by imposing two additional conditions on the BF ansatz. Using the Newman-Penrose formalism, we find that the property of allowing null geodesic shearfree congruences extends beyond type D also to a certain class of metrics of type I .  \nWe then impose a third constraint, ensuring the separability of the Klein-Gordon equation (Section 4). The resulting class of metrics, which we call I B , turns out to be similar to that used by Carter to perform direct integration of Einstein equations in vacuum and electrovacuum gravity [16]. But our ansatz is now valid for type I B and is not constrained by the assumptions about the sources of matter used by Carter. This opens the way to a direct integration of the supergravity equations for this class. Leaving an explicit integration for a separate publication, here we verify that all explicitly known solutions for black holes do admit a simple polynomial description in terms of the constrained BF ansatz.  \nSection 5 is devoted to deriving conditions for the BF metrics to belong to type D. In Section 6 we show that for our class of metrics it is possible to have a unified description of the photon and massive particle surfaces, recently introduced [19, 20] as a tool for analyzing black hole shadows [21, 22]. This follows from a special property of the BF-Killing tensor, which was called slice-reducibility. Our treatment is valid for all basic black holes in extended supergravities. Finally, in Section 7 we present an explicit description of the known black hole solutions in terms of BF functions and give explicit MPS equations for them.', 'II. GENERAL BENENTI-FRANCAVIGLIA ANSATZ': "Our starting point is the class of four-dimensional metrics admitting a pair of commuting Killing vectors. More specifically, we will be interested in the stationary axisymmetric orthogonally transitive spacetimes (SAS) that can be parameterized by a block-diagonal metric, one of whose blocks is spanned by Killing vectors. For parametrization of the SAS spacetime that guarantees the separability of the Hamilton-Jacobi equation we use an ansatz given by Benenti and Francaviglia [18], based on Benenti's theorems [23, 24] proving that in any dimension a necessary and sufficient condition for this to happen is the existence of a commuting system of Schouten-Nijenhuis brackets [25, 26] (for a later discussion, see [27]) for Killing vectors and Killing tensors with a total number equal to the dimension of the spacetime. In this algebra, certain conditions on the eigenvectors of the Killing tensors must also be satisfied.In the fourdimensional case of SAS metrics, to which we restrict ourselves here, one (trivial) Killing tensor is the metric itself, so one irreducible Killing tensor is needed for separability.  \nRecently, SAS spacetimes with prescribed hidden symmetry have attracted attention in the search for viable alternatives to the Kerr metric for astrophysical modeling [28-31]. In these applications, it was found, in particular, that separation of variables in the Klein-Gordon  \nequation for general BF metrics is not guaranteed.  \nA general BF parametrization consists of ten arbitrary functions, each depending on one variable. The metrics are off-shell in the sense that Einstein's equations are not imposed. The SAS metric is written in the coordinates x µ = ( x a , x i ), where x a = t, ϕ correspond to the subspace spanned by the Killing vectors K ( t ) = ∂ t and K ( ϕ ) = ∂ ϕ and x i = r, y , belong to orthogonal two-dimensional space whose metric without loss of generality can be assumed diagonal. BF ansatz looks somewhat simpler in terms of the contravariant metric tensor g µν = ( g ab , g ij ) as follows:  \nwhere two sets of arbitrary functions are introduced A k ( r ) , B k ( y ) , k = 1 .. 5 depending each on one variable, r and y respectively. In order to ensure existence of an exact Killing tensor (EKT), the conformal factor Σ = Σ( r, y ) must be of the special form  \ng ab = Σ -1   A 3 -B 3 A 4 -B 4 A 4 -B 4 A 5 -B 5   , g ij = -Σ -1   A 2 0 0 B 2   (1)  \nΣ = A 1 + B 1 . (2)  \nThen the Killing tensor, satisfying the equation  \n∇ ( α K µν ) = 0 , (3)  \nwhere symmetrization over indices is understood, also has a block diagonal form K µν = ( K ab , K ij ) , where  \nThe inverse metric (1) and the Killing tensor (4) have the following automorphism:  \nΣ K ab =   A 1 B 3 + A 3 B 1 A 1 B 4 + A 4 B 1 A 1 B 4 + A 4 B 1 A 1 B 5 + A 5 B 1   , Σ K ij =   -A 2 B 1 0 0 A 1 B 2   (4)  \nA 1 ↔-B 1 , A i ↔ B i , ( i = 2 .. 4) , g rr ↔-g yy , K rr ↔-K yy . (5)  \nFor an arbitrary conformal factor Σ only a conformal Killing tensor may exist, we will come back to this later. We keep separate notation for the conformal factor for further convenience.  \nTwo blocks for the covariant metric tensor g µν = ( g ab , g ij ): read  \ng ab = Σ P   A 5 -B 5 -A 4 + B 4 -A 4 + B 4 A 3 -B 3   , g ij = -Σ   A -1 2 0 0 B -1 2   (6)  \nwhere  \nP = ( A 3 -B 3 )( A 5 -B 5 ) -( A 4 -B 4 ) 2 . (7)  \nWe shall assume that in a significant region of space-time (e.g. beyond the horizon or the ergosphere) all the BF coefficient functions are positive, and we shall not consider analytic continuation into the negative region, which of course can be done in the usual way. Therefore we shall often use square roots of the BF coefficients, assuming that they are real.  \nOther sign conditions follow from the metric signature with the same reservations:  \nA 3 -B 3 > 0 , A 5 -B 5 < 0 , P < 0 . (8)  \nWe also assume that the conformal factor Σ is positive in the case where it is not assumed to be given by the formula (2).  \nThe static limit corresponds to  \nA 4 ≡ 0 , B 4 ≡ 0 . (9)  \nOne may wonder what is the gauge freedom inside the BF ansatz. The form of the metric suggests possibility of two coordinate transformations  \nr → ˜ r ( r ) , y → ˜ y ( y ) , (10)  \ncontaining two arbitrary functions of independent variables. These can be used to impose additional conditions on A 2 , B 2 , e.g., A 2 = 1 = B 2 (conformally flat metric in the r, y block). Other useful conditions, which turn out to be satisfied by all known supergravity black holes read as follows:  \nA 2 A 5 = a 2 = const , B 2 B 5 = b 2 = const . (11)  \nIn what follows we will assume the validity of this gauge.", 'A. Null tetrad': 'We proceed by choosing some natural Newman-Penrose (NP) null tetrad [32] for the inverse metric tensor(1):  \ng µν = l µ n ν + n µ l ν -m µ ¯ m ν -¯ m µ m ν . (12)  \nWe choose real vectors of the tetrad symmetrically:  \nl = 1 / √ 2Σ ( √ A 3 ∂ t + C ∂ ϕ -√ A 2 ∂ r ) , m = 1 / √ 2Σ ( √ B 3 ∂ t + D∂ ϕ -i √ B 2 ∂ y ) , (13a)  \nwhere  \nn = 1 / √ 2Σ ( √ A 3 ∂ t + C ∂ ϕ + √ A 2 ∂ r ) , ¯ m = 1 / √ 2Σ ( √ B 3 ∂ t + D∂ ϕ + i √ B 2 ∂ y ) . (13b)  \n( A 3 -B 3 ) C = √ A 3 ( A 4 -B 4 )+ √ B 3 √ -P , ( A 3 -B 3 ) D = √ B 3 ( A 4 -B 4 )+ √ A 3 √ -P . (14)  \nRecall that so far the number of arbitrary Benenti functions are ten.', 'III. NULL SHEAR-FREE GEODESIC CONGRUENCES': 'Starting from the general BF class, we would like to select a subclass with additional properties noticed for some supergravity black holes [33]. Vacuum solutions of type D have the property, according to the Goldberg-Sachs theorem [34], of admitting two null geodesic congruences without shear. Electrovacuum black holes arising in pure N = 2 supergravity without matter multiplets are also of type D and also have similar congruences. Recall that stationary charged black holes in N = 2 supergravity can be obtained by Harrison transformations in the three-dimensional sigma model description [5] from the Kerr vacuum metric, so it is likely (although no actual proof has been given) that Harrison transformations preserve this property as well as geodesic integrability. Black hole solutions in extended supergravities with scalar moduli N = 4 , 8, can also be obtained by Harrison transformations from Kerr metric [8, 3537], so we can expect them to share both of the above properties (as mentioned e.g. in [33] for the two-charge STU solution). But Harrison transformations certainly do not preserve the Petrov type of the metric, since supergravity black holes with scalar fields generically belong to type I .  \nThus, first of all we look for a reduced BF ansatz ensuring existence of two null geodesic shear-free congruences.', 'A. First constraints': 'Using algebraic computing and hints from the known supergravity black hole solutions, one is led to consider the following constraints excluding two of ten BF functions:  \nA 4 = √ A 3 A 5 , B 4 = √ B 3 B 5 . (15)  \nWith these conditions the functions entering (14) simplify to  \n√ -P = √ A 3 B 5 -√ A 5 B 3 , ⇒ C = √ A 5 , D = √ B 5 . (16)  \nThis reduced BF ansatz leads to significant simplification of the null tetrad:  \nl = 1 / √ 2Σ ( √ A 3 ∂ t + √ A 5 ∂ ϕ -√ A 2 ∂ r ) , m = 1 / √ 2Σ ( √ B 3 ∂ t + √ B 5 ∂ ϕ -i √ B 2 ∂ y ) , n = 1 / √ 2Σ ( √ A 3 ∂ t + √ A 5 ∂ ϕ + √ A 2 ∂ r ) , ¯ m = 1 / √ 2Σ ( √ B 3 ∂ t + √ B 5 ∂ ϕ + i √ B 2 ∂ y ) . (17)  \nIn the static limit our relations (15) degenerate, so we have to be careful in making correct assignments for the remaining BF coefficients. Looking at the constrained tetrad 17, we realize that a correct choice in the static limit will be  \nA 5 ≡ 0 , B 3 ≡ 0 , A 3 = 0 , B 5 = 0 , (18)  \n/negationslash  \n/negationslash  \nimplying a = 0 , b = 0 .  \n/negationslash', 'B. Newman-Penrose analysis': "To understand which restrictions on the nature of spacetime is put by the first constraints, let's continue our NP analysis. Recall the definitions of the NP projections of the covariant derivatives  \nD = l µ ∇ µ , ∆ = n µ ∇ µ , δ = m µ ∇ µ , ¯ δ = ¯ m µ ∇ µ , (19)  \nand the action of D, ∆ on the vectors l µ , n µ  \nDl µ = ( /epsilon1 +¯ /epsilon1 ) l µ -¯ κm µ -κ ¯ m µ (20)  \n∆ n µ = -( γ + ¯ γ ) n µ + νm µ + ¯ ν ¯ m µ . (21)  \nConsider null congruences aligned with l µ , n µ . If κ = 0 = ν they are geodesic , with /epsilon1, γ being measure of non-affinity. Another important quantity of null congruences is shear , which is defined for them as  \nσ = -m µ δ l µ , ¯ λ = m µ δ n µ (22)  \nrespectively. Calculating the spin coefficients (see Appendix A) for our tetrad, we find:  \nκ = ν = 0 , σ = λ = 0 , (23)  \nwhich means that both congruences are geodesic and shearfree . Other spin coefficients are generically non-zero and pairwise equal:  \nµ = ρ, τ = π, /epsilon1 = γ, α = β. (24)  \nSuch properties are typical for Petrov type D , once two congruences are principal null directions of the Weyl tensor (which is also the case, as we shall see shortly). To establish the Petrov type in our case, we calculate the NP projections of the Weyl tensor:  \nΨ 0 = -C αβγδ l α m β l γ m δ , Ψ 1 = -C αβγδ l α n β l γ m δ , Ψ 2 = -C αβγδ ( l α n β l γ n δ -l α n β m γ ¯ m δ ) / 2 , (25) Ψ 3 = -C αβγδ n α l β n γ m δ , Ψ 4 = -C αβγδ n α ¯ m β n γ ¯ m δ ,  \nFrom the computer assisted calculations one finds that two of them are zero,  \nΨ 0 = 0 = Ψ 4 , (26)  \nwhile the others are rather cumbersome in terms of BF coefficients. Still one can extract the following relation between the other two:  \nΨ 1 = Ψ 3 , (27)  \nreflecting obvious symmetry of the tetrad with respect to interchange A ↔ B . Vanishing of Ψ 0 and Ψ 4 means that the real vectors l µ , n µ are two distinct principal null directions of the Weyl tensor for a constrained BF metric. At the same time, this means that our tetrad is not canonical for determination of the Petrov type. We therefore proceed by computing the values of the quadratic and cubic curvature invariants of the Weyl tensor  \nI = Ψ 0 Ψ 4 -4Ψ 1 Ψ 3 +3Ψ 2 2 , J = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ Ψ 4 Ψ 3 Ψ 2 Ψ 3 Ψ 2 Ψ 1 Ψ 2 Ψ 1 Ψ 0 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ . (28)  \n∣ ∣ As is known, in order for a metric to be algebraically special, the following relationship between these invariants must be satisfied:  \nI 3 = 27 J 2 . (29)  \nIt is easy to see that (27) implies that the constrained BF metrics are algebraically special if in addition to (27) the following conditions hold  \nΨ 2 1 = k Ψ 2 2 , with k = 9 / 16 , or 0 . (30)  \n/negationslash  \nIf (30) does not hold, the metric is of the Petrov type I. If (30) if satisfied with k = 0 and Ψ 2 = 0 then the metric type is D . Other algebraically special types are not possible within the constraint (27), for example if one supposes type II, for which Ψ 0 = 0 = Ψ 1 , one immediately finds that the Weyl tensor completely vanishes, i.e. the metric is of type O .  \nBy Goldberg-Sachs theorem, the vacuum spacetime is algebraically special if it contains a null geodesic shear-free congruence. If there are two such congruences, the Petrov type is D . Therefore in the case of type I spacetime, admitting a null geodesic shear-free congruence, the Ricci tensor should be non-zero. This is the case for supergravity black holes. Thus our class I B consists of non-vacuum metrics, possessing a Killing tensor and a pair of null geodesic shear-free congruences. These properties are close to properties of D type, they will ensure separability of Hamilton-Jacobi equation and Klein-Gordon equation if some further conditions on the Ricci tensor are fulfilled (see below). However a Killing-Yano tensor exists only in the algebraically special case D , so the Dirac equation generally is non-separable.  \nNote that in the null tetrad (17) the vectors l µ , n µ are not affinely parameterized (the spin coefficients /epsilon1, γ being non-zero). One can pass to an affinely parameterized congruence l µ (like in the case of the Kinnersley tetrad for Kerr metric) by rescaling the tetrad vectors as  \nl µ → f -1 ( r, y ) l µ , n µ → f ( r, y ) n µ , f = √ Σ /A 2 . (31)  \nThis leaves unchanged the coefficients (23) and the Weyl projections (27), so our conclusions will be the same.  \nOne can also use the classification scheme based on expansion of the Weyl tensor in terms of bivectors. This leads to the traceless symmetric 3 × 3 matrix Q , related to Weyl projections via  \nQ =      Ψ 2 -1 2 (Ψ 0 +Ψ 4 ) 1 2 i (Ψ 4 -Ψ 0 ) Ψ 1 -Ψ 3 1 2 i (Ψ 4 -Ψ 0 ) Ψ 2 + 1 2 (Ψ 0 +Ψ 4 ) i (Ψ 1 +Ψ 3 ) Ψ 1 -Ψ 3 i (Ψ 1 +Ψ 2 ) -2Ψ 2      . (32)  \nWith our conditions (27) this matrix becomes  \nQ =      Ψ 2 0 0 0 Ψ 2 2 i Ψ 1 0 2 i Ψ 1 -2Ψ 2      . (33)  \nThe case of three different eigenvalues corresponds to type I , one degenerate eigenvalue (in the case Ψ 1 = 0) lead to type D .  \nThus, we have proved that the BF metrics (1) with the additional conditions (15) defines a special Petrov type I B class, which shares with the type D two important properties: 1) it admits two independent null shearfree geodesic congruences, 2) it has an irreducible Killing tensor of the second rank. In terms of the constrained tetrad (17) the Killing tensor has only two non-vanishing Newman-Penrose projections exactly as in the case of the type D metrics:  \nK ln = B 1 , K m ¯ m = A 1 . (34)  \nThis gives some geometric interpretation of two Benenti coefficients entering the conformal factor (2).  \nWe further show that the known solutions for black holes in N = 2 , 4 , 8 extended supergravity theories belong to this class I B , which ensures the separability of the Hamilton-Jacobi equation. Moreover, they satisfy the third constraint, which we are going to establish, ensuring separability of the Klein-Gordon equation.", 'C. Separability of geodesic equations': "Separability of the Hamilton-Jacobi equation  \n∂S ∂x µ ∂S ∂x ν g µν = µ 2 (35)  \ncan be easily demonstrated. Assuming  \nS = -Et + Lϕ + S r ( r ) + S y ( y ) , (36)  \ndenoting p µ = ∂S/∂x µ , and taking into account that all A k depend only on r , while all B k depend only on y , we obtain  \nA 2 p 2 r + U r = 0 , B 2 p 2 y + U y = 0 , (37)  \nU r = C ε 2 a + A 1 µ 2 , U y = -C + ε 2 b + B 1 µ 2 , (38)  \nε a = E √ A 3 -L √ A 5 , ε b = E √ B 3 -L √ B 5 , (39)  \nwhere C is the Carter separation constant related to the Killing tensor as  \nC = p µ p ν K µν = Σ -1 [ A 1 ( ε 2 b + B 2 p 2 y ) + B 1 ( ε 2 a -A 2 p 2 r )] . (40)  \nNote that the above definition of potentials is such that Carter's constant enters additively. This has certain advantages when discussing spherical and conical orbits. The radii of spherical orbits, e.g., correspond to solutions of the equations  \nU r = 0 = U ' r . (41)  \nThe derivative equation does not contain the Carter integral, and thus define radii of spherical orbits as functions of E, L . Then substitution of this radius into the equation U = 0 will show, that the Carter integral for spherical orbits is a function of Killing vector integrals.", 'D. Variable mass': 'In conclusion to this section we explore in which case separability of the Hamilton-Jacobi equation can be extended to the case of variable coordinate-depending mass. This may be of interest in the study of shadows of black holes surrounded by plasma [38-41]. A photon in electron plasma acquires an effective mass determined by the refractive index, which depends on the coordinates through the electron density. Since the latter in the vicinity of black holes depends on the coordinates, the propagation of photons in the geometrical optics approximation is described by the Hamilton-Jacobi equations with mass depending on the coordinates:  \nµ 2 ⇒M 2 ( x ) .  \nSeparation of variables in the Hamilton-Jacobi equation with variable mass is possible only if  \nM 2 ( x ) = M 2 r ( r ) + M 2 y ( y ) A 1 + B 1 . (42)  \nThis is consistent with what was found for the Kerr metric in [41].', 'IV. SEPARABILITY OF KLEIN-GORDON EQUATION': "Generic type I B class of metrics still does not quarantee separability of the wave equations. Consider the Klein-Gordon equation for a real scalar field φ :  \n/square φ = 1 √ -g ∂ µ ( √ -gg µν ∂ ν φ ) = -µ 2 φ. (43)  \nThe metric determinant is crucial for separability. It is easier to calculate it in BF form using contravariant components (1). One obtains  \n√ -g = Σ 2 √ A 2 B 2 √ -P (44)  \nWith the first two constraints (15), one can use (7) to obtain P , leading to  \n√ -g = Σ 2 √ A 2 B 2 ( √ A 3 B 5 -√ A 5 B 3 ) . (45)  \nConsidering the inverse metric (1), it becomes clear that the condition for separability is  \n√ -g = Σ .  \nThis leads to the third constraint on the Benenti coefficient functions  \nΣ = √ A 2 B 2 ( √ A 3 B 5 -√ B 3 A 5 ) , (46)  \nwhich can also be rewritten as  \nA 1 + B 1 = bA 23 -aB 23 , (47)  \nwhere we introduced  \nA 23 = √ A 2 A 3 , B 23 = √ B 2 B 3 , (48)  \nand used the gauge (11). Differentiating this with respect to the relevant arguments, one can find useful differential relations following from the third constraint:  \nA ' 1 = bA ' 23 , B ' 1 = aB ' 23 . (49)  \n(Note that primes cannot be omitted in these ratios!) With the help of (44) it is easy to establish the separability of equation (43). Substituting the product  \nφ ( x µ ) = e -iωt + imϕ R ( r ) Y ( y ) , (50)  \nand dividing the Eq.(43) by φ we obtain:  \n(( A 2 ) ' R ) ' R + (( B 2 ) ' Y ) ' Y + U ( r ) -V ( y ) = 0 , (51)  \nwhere the primes denote derivatives with respect to the corresponding arguments r, y , and the potential terms are equal to  \nU ( r ) = ( ω √ A 3 -m √ A 5 ) 2 -µ 2 A 1 , (52)  \nV ( y ) = ( ω √ B 3 -m √ B 5 ) 2 + µ 2 B 1 . (53)  \nThe separability of Eq. (51) is obvious. The third constraint effectively reduces the number of arbitrary functions to seven, two of which A 2 , B 2 can still be fixed using the gauge freedom, so the number of essentially independent functions is five.  \nThe separability of the Klein-Gordon equation can also be investigated using the secondorder Carter differential operator associated with the Killing tensor:  \nˆ K = ∇ µ K µν ∇ ν , (54)  \nwhich must commute with D'Alembert operator [33]. The commutator was elaborated in [42]:  \n[ /square , ˆ K ] φ = 4 3 ∇ α ( K σ [ α R β ] σ ) ∇ β φ. (55)  \nProjecting the tensor on the right side onto the NP tetrad, we obtain  \nK σ [ α R β ] σ = 2( K ln + K m ¯ m )( n β ( ¯ m α Φ 01 + m α Φ 10 ) -n α ( ¯ m β Φ 01 + m β Φ 10 )+ +( l β ¯ m α -l α ¯ m β )Φ 12 +( l β m α -l α m β )Φ 21 ) . (56)  \nThus, a sufficient condition for commutativity is that two Ricci scalars vanish:  \nΦ 01 = R µν l µ m ν / 2 = Φ 10 , Φ 12 = R µν n µ m ν / 2 = Φ 21 . (57)  \nTheir expression, taking into account the third constraint (46) in terms of Benenti quantities, is given by formula (B2) in Appendix B. Equating them to zero, we obtain:  \naA '' 23 -bB '' 23 = 0 , (58)  \nwhere the primes denote derivatives with respect to the corresponding arguments. Given (49), this can also be rewritten as  \na 2 A '' 1 -b 2 B '' 1 = 0 . (59)  \nSince one term is a function of r and the other is a function of y , each must be a constant. In other words, A 1 and B 1 must be at most quadratic polynomials of the corresponding arguments.  \nLet us give an NP-description of the operators /square and ˆ K in terms of derivatives (19) and the spin coefficients:  \n/square = ( D + ε + ¯ ε -ρ -¯ ρ )∆ + (∆ + µ + ¯ µ -γ -¯ γ ) D --( δ -τ -¯ α + β + ¯ π ) ¯ δ -( ¯ δ -α -¯ τ + π + ¯ β ) δ, (60)  \nˆ K = ( D + ε + ¯ ε -ρ -¯ ρ ) K ln ∆+(∆+ µ + ¯ µ -γ -¯ γ ) K ln D + +( δ -τ -¯ α + β + ¯ π ) K m ¯ m ¯ δ +( ¯ δ -α -¯ τ + π + ¯ β ) K m ¯ m δ, (61)  \nIt is useful to introduce the following set of operators:  \nD ± n = √ A 3 /A 2 ∂ t ± ∂ r + √ A 5 /A 2 ∂ ϕ ± n∂ r (ln A 2 ) , (62a)  \nwhere n and s are integers. These operators satisfy two simple identities:  \nL ± s = √ B 3 /B 2 ∂ t ± i ∂ y + √ B 5 /B 2 ∂ ϕ ± is ∂ y (ln B 2 ) , (62b)  \nD ± n A 2 k = A 2 k D ± n + k , L ± s B 2 l = B 2 l L ± s + l . (63)  \nIn terms of these operators, the directional derivatives acting on a scalar function will have the form:  \nD = √ A 2 / 2Σ D -0 , ∆ = √ A 2 / 2Σ D + 0 , δ = √ B 2 / 2Σ L -0 , ¯ δ = √ B 2 / 2Σ L + 0 . (64)  \nAlso, taking into account the explicit form of the spin coefficients (A1-A4), we can obtain the following representations for the combinations included in the d'Alembertian: Also, taking into account the explicit form of the spin coefficients (A1-A4), we can obtain the following representations for the combinations included in the d'Alembertian:  \nε + ¯ ε -ρ -¯ ρ = √ A 2 / 8Σ 3 ( 2Σ ∂ r ln [ √ A 3 B 5 -√ A 5 B 3 ] -3 ∂ r Σ ) , (65a)  \nIt is easy to see that in the general case, after substitution into the Klein-Gordon equation, the latter does not allow separation of variables unless the second constraint (46) is imposed. In this case, our expressions are simplified to  \n¯ π -τ + β -¯ α = -i √ B 2 / 8Σ 3 ( 2Σ ∂ y ln [ √ A 3 B 5 -√ A 5 B 3 ] -3 ∂ y Σ ) . (65b)  \nε + ¯ ε -ρ -¯ ρ = √ A 2 ∂ r (2Σ) -1 / 2 -(2Σ) -1 / 2 ∂ r √ A 2 , (66a)  \nUsing Eqs. (62-66), the d'Alembert operator (60) can be cast into the form  \n¯ π -τ + β -¯ α = i √ B 2 ∂ y (2Σ) -1 / 2 -i (2Σ) -1 / 2 ∂ r √ B 2 . (66b)  \n/square = 1 2Σ [ A 2 ( D -1 D + 0 + D + 1 D -0 ) -B 2 ( L -1 L + 0 + L + 1 L -0 )] , (67)  \nwhile the Carter operator can be presented as  \nˆ K = 1 2Σ [ A 2 B 1 ( D -1 D + 0 + D + 1 D -0 ) + B 2 A 1 ( L -1 L + 0 + L + 1 L -0 )] = B 2 2 ( L -1 L + 0 + L + 1 L -0 ) + B 1 /square . (68)  \nBy direct substitution one can verify that it commutes with the d'Alembertian (67), taking into account the fact that the second term in the solution space /square φ = 0 vanishes.", 'A. Separable supergravity backgroounds': "By imposing three constraints on the BF coefficients, we arrive at the following metric paremeterization:  \nds 2 = A 2 B 2 Σ ( √ B 5 dt -√ B 3 dϕ ) 2 -A 2 B 2 Σ ( √ A 5 dt -√ A 3 dϕ ) 2 -Σ A 2 dr 2 -Σ B 2 dy 2 , (69)  \nΣ = √ A 2 B 2 ( √ A 3 B 5 -√ A 5 B 3 ) . (70)  \nwhere  \nThis is precisely the Carter metric ansatz [16] for which Einstein's vacuum and electrovacuum equations were solved directly. Several families of solutions were obtained among which Kerr and Kerr-Newman black holes belonging to the Petrov type D (for further applications see also [17])  \nConsider now the generic 4D supergravity bosonic action which is a special scalar-vectortensor theory with multiple scalar and vector fields [5, 6]. It includes n s scalar moduli Ψ A , A = 1 , . . . , n s (dilatons and axions) and n v abelian vector fields F I = dA I , I = 1 , . . . , n v :  \nS = ∫ d 4 x [( R -1 2 f AB ∂ µ Ψ A ∂ µ Ψ B -1 2 K IJ F I µν F Jµν ) √ -g -1 2 H IJ F I µν F J λτ /epsilon1 µνλτ ] . (71)  \nThe scalar moduli parametrize a four-dimensional coset (e.g. U (8) /E 7(7) for N = 8 supergravity) with an associated target metric f AB . Vector fields transform under the same global symmetry implemented by real symmetric matrices K IJ , H IJ depending on scalar fields Ψ A (summation over the repeated indices I, J is understood). The corresponding Einstein equations read:  \nR µν = 1 2 f AB Ψ A ,µ Ψ B ,ν -K IJ ( F I µλ F Jλ ν + 1 4 g µν F I αβ F Jαβ ) . (72)  \nThe scalar fields depend only on r, y , so they contribute directly only in transverse part of the Ricci tensor. The Maxwell sector at the right hand side of this equations has the same structure as in pure Einstein-Maxwell system. All this ensure (the details will be given elsewhere) that supergravity Einstein equations will separate for our metric ansatz similarly to Einstein-Maxwell case [16]. Note, that the Smarr mass formulas for black hole solutions of the theory (71) was shown recently to repeat the case of the Einstein-Maxwell theory [43].  \nTo our knowledge, no direct integration of the Einstein supergravity equations by separation of variables has been performed so far, although the leading solutions for black holes have been obtained using indirect methods. In addition to Harrison transformations, one can mention the guesswork of obtaining the BPS solution [9] or the integration of the null geodesic equations for  \nthe target space of the sigma model [14] (also known as the nilpotent orbit method [15]). The situation is different in the electrovacuum case, where large classes of solutions were obtained by direct integration of the Ernst equations.", 'V. KILLING-YANO AND TYPE D': 'Here we find conditions on the BF functions that guarantee that the solution is algebraically special, which for our class I B means type D . In this case, the Killing-Yano tensor that guarantees the separability of the Dirac equation also exists. We will use this as a tool to find conditions for type D .', 'A. Killing-Yano': "The Killing-Yano tensor Y µν = -Y νµ satisfying the equation  \n∇ ( α Y µ ) ν = 0 , (73)  \ncan be regarded as a 'square root' of the Killing tensor:  \nY µ α Y αν = K µν . (74)  \nSince we know the Killing tensor (4) independently of Petrov type of the metric, we may consider Eqs.(73) and (74) as independent conditions which prescribe the metric to be of type D , and define the KY tensor itself. In NP description, our Killing tensor has only two non-zero components (34): K ln and K m ¯ m . So projecting (74) on the NP tetrad, one obtains:  \nY 2 ln = K ln , Y 2 m ¯ m = -K m ¯ m , (75)  \nthe other NP components of the KJ tensor being zero. Together with Eq. (34) this gives  \nY ln = p √ B 1 , Y m ¯ m = iq √ A 1 , p = ± 1 , q = ± 1 , (76)  \nwhere we introduced sign factors to be fixed later.", 'B. Consistency conditions for type D': "Now we have to satisfy the KY equation (73) for consistency, in other words, we have to find new constraint equations on the BF coefficient functions which ensure that solution belongs to  \ntype D . Projecting KJ equation (73) onto the NP tetrad we get sixteen equations, from which, with account for (34) and pairwise equality of the spin coefficients (24), only the following six are relevant:  \n( τ + ¯ π ) Y ln -( τ -¯ π ) Y m ¯ m = 0 , ( ρ + ¯ ρ ) Y ln -( ρ -¯ ρ ) Y m ¯ m = 0 , (77a)  \nDY m ¯ m -ρ ( Y ln -Y m ¯ m ) = 0 , δY ln -¯ π ( Y ln + Y m ¯ m ) = 0 , (77b)  \n∆ Y m ¯ m + µ ( Y ln -Y m ¯ m ) = 0 , δY ln + τ ( Y ln -Y m ¯ m ) = 0 . (77c)  \nFor the spin coefficients involved, from (A1,A2) with account for the constraint (46) one obtains:  \nµ = ρ = b 2Σ √ A 2 / 2Σ( A ' 23 -i B ' 23 ) , τ = π = a 2Σ √ B 2 / 2Σ( A ' 23 -i B ' 23 ) , (78)  \nThen from the first pair of equations of the system we get the following relation:  \np A ' 23 √ B 1 = q B ' 23 √ A 1 , (79)  \nwhich separates into pair of one-variable equations:  \npA ' 23 = 2 √ A 1 , qB ' 23 = 2 √ B 1 (80)  \nwhere we introduced the coefficient two for further convenience. Now we have to satisfy the second and third pairs of equations (77b, 77c). Taking into account definition of NP derivatives (19), we obtain  \n( √ A 1 ) ' = b/p, ( √ B 1 ) ' = -a/q (81)  \nFrom (2) and (46) follows that  \nA 1 + B 1 = bA 23 -aB 23 .  \nTo satisfy this, one has to choose p = 1 and q = -1, Thus, we have found a system of restrictions on the BF coefficients that guarantee the existence of the KY tensors, i.e. the belonging of the metric to the D type:  \nA ' 1 = 2 b √ A 1 , B ' 1 = 2 a √ B 1 ; (82a)  \nIntergration of this system provides generic form for some of the metric coefficients for type D BF sector:  \nA ' 23 = 2 √ A 1 , B ' 23 = -2 √ B 1 . (82b)  \nA 1 = ( br + c 1 ) 2 , B 1 = ( ay + d 1 ) 2 , (83a)  \nA 23 = br 2 +2 c 1 r + c 2 , B 23 = -( ay 2 +2 d 1 y + d 2 ) , (83b)  \nwhere the constants c 1 , d 1 , c 2 and d 2 according to equations (2, 46) are subject to the following condition:  \nb c 2 + ad 2 = c 1 2 + d 1 2 . (84)  \nOnly when all these conditions are satisfied does the Killing-Yano tensor become legitimate and finally given by the NP components  \nY ln = √ B 1 , Y m ¯ m = -i √ A 1 . (85)  \nIt remains to check that with these conditions the Weyl tensor projections Ψ 1 , 3 vanish. Taking into account the conditions (11 15, 46), Ψ 1 , 3 can be written in the form  \n8Σ 3 Ψ 1 , 3 = -√ A 2 B 2 { Σ( aA '' 23 -bB '' 23 ) -ab ( A ' 23 2 + B ' 23 2 ) } , (86)  \nSubstituting here (83a 83b 84), one finds that Ψ 1 , 3 = 0 indeed. Also note that in the static case, when A 5 = 0 = B 3 and hence a = 0 , B 23 = 0, we have Ψ 1 , 3 = 0 without imposing a condition of type D .  \nThe Ricci scalars and the Weyl scalar Ψ 2 for generic type I B are given in the Appendix B.", 'C. Conformal Killing tensor': 'The conformal Killing tensor (CKT) must satisfy the equations  \n∇ ( α K µν ) = Ω ( α g µν ) , 6 Ω α = (2 ∇ σ K α σ + ∇ α K ) , K = g µν K µν . (87)  \nFor Ω α = 0, these equations in our I B class of metrics have a solution with only two NP projections K ln , K m ¯ m , so it is natural to look for CKT of similar structure. Projecting Eq. (87) onto the chosen tetrad (17) one gets the following system of four equations  \n( D + ρ + ¯ ρ )( K ln + K m ¯ m ) = 0 , ( δ + τ -¯ π )( K ln + K m ¯ m ) = 0 , (∆ -µ -¯ µ )( K ln + K m ¯ m ) = 0 , ( ¯ δ + ¯ τ -π )( K ln + K m ¯ m ) = 0 . (88)  \nAfter substitution of an explicit expression (A1, A2) for the spin coefficients, this can be easily integrated. Thus, up to an arbitrary function S ( r, y ), we can write down the NP projections  \nK ln = Σ -S ( r, y ) , K m ¯ m = S ( r, y ) . (89)  \nThe conformal factor Σ = Σ( r, y ) here may be non-separable. The conformal tensor transforms into an exact one for S = A 1 and the condition (2).', 'A. Black hole shadows and characteristic surfaces': 'Recent interest to black hole shadows [21, 22] gave rise to new theoretical approaches. Strong gravitational lensing, quasi-normal modes of black holes and the formation of black hole shadows are determined by the motion of massless particles near photon surfaces [44-46], on which photons winds before scatter to infinity. Such surfaces exist in static metrics, while in rotating case similar role is played by surfaces where non-planar spherical orbits are located [47]. In the first case, the corresponding hypersurfaces in space-time are umbilic [48] (the tensor of external curvature is proportional to the induced metric), in the second case they are partially umbilic, which means that the latter property is satisfied not for tensors as a whole, but by their convolutions with a part of the vectors of the tangent space.  \nIn a similar way, one can consider the characteristic surfaces of massive particles, as well as particles of variable mass, for example, photons in plasma [49, 50]. It was noted [51-53] that the existence of hypersurfaces with the above properties correlates with the existence of the Killing tensor. Eventually, it was shown [19, 20] that Killing tensors, which reduce to trivial (products of Killing vectors) on hypersurfaces that can be used to stratify the entire spacetime, ensure that these hypersurfaces contain generalized photon surfaces, including those associated with the motion of massive particles.  \nIt is remarkable that the BF Killing tensor (4) possesses the slice-reducibility property. As a result, the BF ansatz can be used to give unified description of particle surfaces and shadows of supergravity black holes [19, 20, 49, 50].', 'B. Slice-reducibility': 'To see that Benenti Killing tensor (4) is slice-reducible [20], it is enough write it in the form  \nK µν = -A 1 g µν -A 2 δ µ r δ ν r + ˜ K µν r , (90)  \nwhere  \n˜ K µν r = A 3 δ µ t δ ν t +2 A 4 δ ( µ t δ ν ) ϕ + A 5 δ µ ϕ δ ν ϕ . (91)  \nThe first term in (90) is trivial Killing tensor on S r , since A 1 = const there. The second term is orthogonal to S r and thus irrelevant, while the third term ˜ K µν r is a reducible Killing tensor  \non this hypersurface, being presented as linear combination of the tensor products of Killing vectors projected onto it.  \nSimilarly the BF Killing tensor can be presented in terms of S y foliation:  \nK µν = B 1 g µν + B 2 δ µ y δ ν y + ˜ K µν y . (92)  \nwith the slice projection  \n˜ K µν y = B 3 δ µ t δ ν t +2 B 4 δ ( µ t δ ν ) ϕ + B 5 δ µ ϕ δ ν ϕ . (93)  \nIf we omit terms with the metric tensor in Eqs. (90) and (92), we get simple expressions for conformal Killing tensors, which depend only on one coordinate.  \nThus, the Benenti ansatz ensures the Killing tensor is slice-reducible with respect to both foliations of spacetime. In particular, automorphism (5) naturally arises as a symmetry of slices discussed in [20].', 'C. Photon and massive particle surfaces': "The black hole horizon r h is the largest root of the equation  \nA 2 ( r h ) = 0 , A 2 > 0 for r > r h . (94)  \nIn spacetime this is a null hypersurface. Consider the timelike three-dimensional hypersurface S r for r = const > r h with the unit outward normal spacelike covector n µ = -√ Σ /A 2 δ r µ . The induced metric and the extrincsic curvature of S r in the bulk coordinates read  \nh µν = g µν + n µ n ν , χ µν = h α µ h β ν ∇ α n β = h α µ h β ν ( n β,α -Γ λ αβ n λ ) . (95)  \nχ µν dx µ dx ν = n r 2 g rr ( g ab,r dx a dx b + g yy,r dy 2 ) (96)  \nFrom here one finds:  \nThe hypersurface S r in the static spacetime contains a photon surface, like r = 3 M in Schwarzschild metric, which is the loci of confined photon orbits. Its radius is determined by the umbilicity condition  \nχ µν = χ α α 3 h µν , µ, ν = t, ϕ, y (97)  \n(ln g tt ) ,r = (ln | g ϕϕ | ) ,r = (ln | g yy | ) ,r (98)  \nwhich reduces to  \nSubstituting BF coefficients one finds that the last equality is trivially satisfied (showing that we are dealing with geometrical sphere), while the first one defines the radius of the photon sphere. It may be written as  \n(ln A 2 ) ' = 2(ln A 23 ) ' , or simply A ' 3 = 0 , (99)  \nwhere prime denotes the derivative with respect to r . Let's test this equation for the ReissnerNordstrom metric, in which case A 2 = r 2 -2 Mr + Q 2 , A 23 = r 2 . From Eq. (99) one obtains  \nr 2 -3 Mr +2 Q 2 = 0 ⇒ r = 3 2 ( M ± √ M 2 -8 Q 2 / 9 ) , (100)  \nwhich is indeed a correct expression for the photon spheres in Reissber-Nordstrom spacetime. One can also look for timelike hypersurfaces such that a particle of mass µ moving along an initially tangent worldline to S r with tangential vector p µ (normalized as p µ p ν = µ 2 ) remains there forever. Such a surface was named massive particle surface (MPS). The surface radius r will depend on the particle energy and angular momentum integrals E, L , and the family of MPS will foliate certain four-dimensional subspace in space-time. The (partial) umbilicity condition in this case must be satisfied only in those directions of the tangent space of S r that are orthogonal to the Killing vectors.  \nIn the general stationary case, the induced metric and the second quadratic form can be written as  \nh µν dx µ dx ν = A 2 Σ ( bdt -B 23 dϕ ) 2 -B 2 Σ ( adt -A 23 dϕ ) 2 -Σ B 2 dy 2 , (101)  \nχ µν dx µ dx ν = 1 2 √ A 2 Σ ∂ r h µν dx µ dx ν . (102)  \nIn this case the MPS radius is a solution of the following equation of partial umbilicity [49]:  \n1 2 h yy √ | g rr | ∂ r p 2 = χ yy ( p 2 -µ 2 ) , p 2 = g ab p a p b , p a = ( -E,L ) , (103)  \nwhere E, L are constants of motion defined in (36). Substition of BF coefficient functions leads to  \n( √ A 3 E -√ A 5 L ) ' ( √ A 3 E -√ A 5 L ) = 1 2 µ 2 A ' 1 (104)  \nwhere primes denotes derivatives over r . This equation is nothing but U ' r = 0 for the radial potential defined in (38). As was noted in Sec. 3, for spherical orbits the Carter constant is determined by two other integrals of motion. This explains why the equation for MPS radius defines it as function of two parameters only.  \nDetailed description of spherical orbits of massive particles in Kerr spacetime was given by Teo, [47], where the suitable domains of C are determined. More general setting, including the case of charged particles and presence of the electromagnetic field can be found in [49]. In the massless case, spherical orbits are determined by the ratio L/E , called the impact parameter, due to the scale invariance of the geodesic equations.  \nOur Eq. (104) is a general equation defining the MPS radius in terms of the motion integrals E, L for the entire class of I B -type metrics. The regions of these parameters is restricted by consistency conditions, this is discussed in detail in [47, 49]. The particular cases are described in the next section.", 'VII. SUPERGRAVITY BLACK HOLES': 'Black hole solutions in N = 2 , 4 , 8 supergravity models have been constructed using the Harrison transformations applied to the Kerr metric. The most general results were obtained in this way by Chow and Compere [37]. Here we show that the known solutions indeed belong to our doubly restricted (15,46) BF class I B . This class ensures the separability of not only the Hamilton-Jacobi and Klein-Gordon equations, but also the Einstein equations [16]. All metrics also satisfy the gauge condition (11) in Boyer-Lindquist coordinates with the gauge constant a equal to the rotation parameter. Therefore, in this section we will use the a symbol in both senses.', 'A. Kerr-Newman': 'Kerr-Newman metric may be viewed as type D solution of N = 2 pure supergravity which does not involve scalar fields. In Boyer-Lindquist coordinates, the interval reads:  \nds 2 = ∆ r / Σ ( dt -a sin 2 θ dϕ ) 2 -1 / Σ sin 2 θ ( adt -( r 2 + a 2 ) dϕ ) 2 -Σ / ∆ r dr 2 -Σ dθ 2 , ∆ r = r 2 -2 Mr + a 2 + Q 2 , Σ = r 2 + a 2 cos θ,  \nThe corresponding BF metric coefficients A k ( r ) and B k ( y ) read:  \nA 1 = r 2 , B 1 = a 2 y 2 A 3 = ( r 2 + a 2 ) 2 / ∆ r , B 3 = a 2 (1 -y 2 ) , A 5 = a 2 / A 2 = a 2 / ∆ r , B 5 = 1 / B 2 = 1 / (1 -y 2 ) , (105)  \nwhere y = cos θ . It is easy to verify that conditions (15,46) are fulfilled:  \nA 2 4 = A 3 A 5 , B 2 4 = B 3 B 5 , b = 1 , A 1 + B 1 = √ A 2 B 2 ( √ A 3 B 5 -√ A 5 B 3 ) . (106)  \nThe BF formulas for type D (83a,83b,84) also hold. The non vanishing Weyl and Ricci scalars for this metric are  \nΨ 2 = Q 2 -M ( r -i ay ) ( r -i ay ) 3 ( r + i ay ) , Φ 11 = Q 2 2( r 2 + a 2 ) 2 . (107)  \nThe MPS equation is a 5-th order polynomial  \n( E ( r 2 + a 2 ) -aL ) [ Er (∆ r -Mr + Q 2 ) + aL ( r -m ) + a 2 EM ] -µ 2 r ∆ r 2 = 0 , (108)  \nwhich can be subjected to numerical analysis.', "B. Gal'tsov-Kechkin solution": "The seven-parametric family of rotating dilaton-axion-NUT dyons in truncated N = 4 supergravity obtained in [8] reads:  \nds 2 = ∆ r -a 2 sin 2 θ Σ ( dt -wdϕ ) 2 -Σ ( dr 2 ∆ r + dθ 2 + ∆ r sin 2 θ ∆ r -a 2 sin 2 θ dϕ 2 ) , ∆ r = ( r -r -)( r -2 M ) + a 2 -( N -N -) 2 , Σ = r ( r -r -) + ( a cos θ + N ) 2 -N 2 -, w = 2 a 2 sin 2 θ -∆ r [ N ∆ r cos θ + a sin 2 θ ( M ( r -r -) + N ( N -N -)) ] , r -= M | Q -iP | 2 M + iN 2 , N -= N | Q -iP | 2 2 M + iN 2 .  \n| | | |  \nThe physical parameters are mass M , NUT-charge N , electric and magnetic charges Q, P , a rotation parameter a , the asymptotic values of dilaton and axion a set zero (for P = 0 = N this metric transforms to the Sen metric [54] via the coordinate shift r → r + r -). In the BF form (1) with cos θ = y we have:  \nA 1 = r ( r -r -) , B 1 = ( ay + N ) 2 -N 2 -, A 3 = ( r ( r -r -) + a 2 + N 2 -N 2 -) 2 / ∆ r , B 3 = [ a (1 -y 2 ) -2 Ny ] 2 / 1 -y 2 , A 5 = a 2 / A 2 = a 2 / ∆ r , B 5 = 1 / B 2 = 1 / 1 -y 2 , (110) A 2 4 = A 3 A 5 , B 2 4 = B 3 B 5 , a = a, b = 1 , A 1 + B 1 = √ A 2 B 2 ( √ A 3 B 5 -√ A 5 B 3 ) ,  \n(109)  \nthe last relation being the separability condition (46) for the Klein-Gordon equation. The metric is a non-vacuum Petrov type I B solution, with the following set of non-zero Weyl and Ricci scalars:  \nΨ 1 = Ψ 3 = a (4 N 2 -+ r 2 -) sin θ √ ∆ r 8Σ 3 ,  \n12Σ 3 Ψ 2 = -12( M -iN )( r + i ( a cos θ + N )) 3 +6 NN -( 2( r + i ( N + a cos θ )) -r -) 2 + + r 3 -(8 M -r ) + 8 NN 3 --4 N 4 --6 ar -cos θ (5 M -3 iN )( a cos θ +2 N -2 ir )+ +2 r -( M ( 15( r + iN ) 2 +7 N 2 -) +18 N 2 r +9 iN 3 +3 iN ( N 2 --3 r 2 ) -2 N 2 -r ) + +4 N 2 -( 2 a 2 -a cos θ (3 i ( M + iN ) + a cos θ ) -5 Mr +2 N 2 -3 iN ( M + r ) ) + +4 N 2 -r 2 + r 2 -( 2 a 2 -24 iMN -7 N 2 +2 NN --N 2 -) --r 2 -( a cos θ (6 i (4 M -iN ) + a cos θ ) + 26 Mr -6 iNr -r 2 ) ,  \n16Σ 3 Φ 11 = a 2 cos 2 ( θ ) ( 8 Mr -+16 NN --4 N 2 --r 2 -) +16 MN 2 -r --24 MN 2 -r + +8 aN cos θ ( 2 Mr -+4 NN --4 N 2 --r 2 -) + N 2 ( 8 Mr --28 N 2 --7 r 2 -) + +8 Mr -r 2 -14 Mr 2 -r +16 N 3 N -+2 NN -( 4 N 2 -+3 r 2 -+8 r 2 -8 r -r ) + + N 2 -r 2 --4 N 2 -r 2 +4 N 2 -r -r +4 N 4 --r 2 -r 2 + r 3 -r +6 Mr 3 -, Λ = R 24 = -( 4 N 2 -+ r 2 -) ( a 2 sin 2 θ +∆ r ) 48Σ 3 . (111)  \nMatter fields ensure that the Carter and d'Alembert operators commute, since the NPprojectors of the Ricci tensor 57 are zero. The MPS equation reads:  \nE ( r ( r -r -) + N 2 -N 2 -) [ (2 r -r -)( r 2 -2 rr -+ a 2 + N 2 -2 NN -+3 N 2 -) --2 M (3 r 2 -rr -+2 r 2 -+ N 2 -N 2 -) ] -4 a 2 ELM + a 4 E 2 (2 r +2 M -r -)+ + a 2 [ 2 E 2 (2(2 r -r -)( N 2 --NN -) -2 M (( r -r -) 2 ) -N 2 + N 2 -) -L 2 (2 r -r --2 M ) ] + +4 aEL [ M ( r -r -) -N 2 + N 2 -+ N -( N -N -)(2 r -r -) ] -µ 2 ∆ 2 (2 r -r -) = 0 .", 'C. SWIP solutions': "Another family of supersymmetric extremal stationary solutions of N = 4 , D = 4 supergravity containing a set of electric and magnetic charges satisfying Bogomol'nyi (BPS) bound  \nwas obtained in [9]. The metric has the same form (109) with  \n2  \nw = ∆ r -a 2 sin 2 θ ( N ∆ r cos θ + a sin 2 θ [ m ( r -( m + | Υ | )) + 1 2 ( |M| 2 -| Υ | 2 ) ]) , ∆ r = r [ r -2( m + | Υ | )] + a 2 +( m + | Υ | ) 2 , Σ = r ( r -2 | Υ | ) + ( a cos θ + N ) 2 . (112)  \nThe solution depends on complex mass M = m + iN , axion-dilaton charge Υ = -2 M ∑ n [ ¯ Γ ( n ) ] 2 , and electromagnetic charges Γ ( n ) = 1 2 ( Q ( n ) + iP ( n ) ) , n = 1 , 2. The BPS identity reads:  \n|M| 2 + | Υ | 2 -4 ∑ n | Γ ( n ) | 2 = 0 . (113)  \nIn this case  \nA 1 = r ( r -2 | Υ | ) , B 1 = N + ay, A 3 = ( r ( r -2 | Υ | ) + N 2 + a 2 ) 2 / ∆ r , B 3 = ( a (1 -y 2 ) -2 Ny ) 2 / 1 -y 2 , A 5 = a 2 / A 2 = a 2 / ∆ r , B 5 = 1 / B 2 = 1 / (1 -y 2 ) . (114)  \nThe non-zero NP quantities are:  \n2 √  \nΨ 1 = Ψ 3 = a sin θ | Υ | ∆ r 2Σ 3 , 3Σ 3 Ψ 2 = 3( m -iN )( m -r -ia cos θ )( r + i ( N + a cos θ )) 2 + +3( m -iN )(3( r + ia cos θ ) -2 m + iN )( r + i ( N + a cos θ )) | Υ |--2 | Υ | 2 (3 N 2 -2( m 2 + r 2 ) + 7 mr +6 iN ( m -r ) + a 2 cos 2 θ ) --2 | Υ | 3 (4( r -m ) + 3 i ( N + a cos θ )) + 3 ia | Υ | 2 cos θ (2( m -iN ) -r ) + 4 | Υ | 4 , 4Σ 3 Φ 11 = 3 | Υ | 2 ( a 2 cos 2 θ +( m -r + | Υ | ) 2 )+ +2Σ( m 2 -a cos θ (2 N + a cos θ ) -( r -| Υ | ) 2 +Σ) , Φ 00 = Φ 22 = | Υ | 2 ∆ r 2Σ 3 , Φ 02 = -a 2 | Υ | 2 sin 2 θ 2Σ 3 , Λ = -| Υ | ( a 2 (1 + sin 2 θ ) + ( m -r + | Υ | ) 2 ) 12Σ 3 . (115)  \nSince Φ 01 = Φ 12 = 0, the separability condition for the Klein-Gordon equation aldo holds.  \nThe MPS equation reads:  \nE ( r ( r -2 | Υ | ) + N 2 + a 2 ) [ { E (∆ r + | Υ | 2 -M 2 -N 2 ) + aL } ( r -M -| Υ | ) + 2 a 2 EM ] --µ 2 ∆ 2 r ( r -| Υ | ) = 0 . (116)", 'D. STU black holes': 'The STU solution is a general asymptotically flat, stationary black hole in supergravity N = 8 [37], parameterized by mass M , rotation parameter a , and four electric charges s I , I = 1 , .., 4. The metric has the same form (109) with  \n∆ r = r 2 -2 Mr + a 2 , w = -2 Maω sin 2 θ ∆ r -a 2 sin 2 θ , ω = ((Π c -Π s ) r +2 M Π s ) , Σ 2 = 4 ∏ I =0 ( r +2 Ms 2 I ) + a 4 cos 4 θ +  \n+2 a 2 cos 2 θ ( r 2 + Mr 3 ∑ I =0 s 2 I +4 M 2 (Π c -Π s )Π s -2 M 2 3 ∑ I<J<K s 2 I s 2 J s 2 K ) ,  \nand the products of charges are defined as follows  \nΠ s = 4 ∏ i =1 s I = 4 ∏ I =1 sinh δ I , Π c = 4 ∏ I =1 √ 1 + s 2 I = 4 ∏ I =1 cosh δ I , s 2 I = sinh 2 δ I .  \nIn the case of pairwise equality of charges s 1 = s 3 = S 1 , s 2 = s 4 = S 2 we are dealing with the so-called two-charge solution. If S 2 = 0, then the metric reduces to the Kerr-Sen solution [54]. For a more general classification, see [35]. It is possible to transform the metric to the BF form (1) only in the case of a two-charge solution, for which  \nΣ 2 ch = r 2 + a 2 cos 2 θ +2 Mr ( S 2 1 + S 2 2 ) + 4 M 2 S 2 1 S 2 2 = r 2 -2 Mr + a 2 cos 2 θ +2 Mw 2 ch (117)  \nThen for the functions A i ( r ), B i ( x ) we have:  \n2 2 2  \nA 1 = r -2 Mr +2 Mw 2 ch , B 1 = a y , A 3 = ( r 2 -2 Mr + a 2 +2 Mw 2 ch ) 2 / ∆ r , B 3 = a 2 (1 -y 2 ) , A 5 = a 2 / A 2 = a 2 / ∆ r , B 5 = 1 / B 2 = 1 / (1 -y 2 ) . (118)  \nThe Weyl and Ricci scalars are  \n4Σ  \nΨ 1 = Ψ 3 = aM 2 sin θ ( S 2 1 -S 2 2 ) 2 √ ∆ r 2Σ 3 2 ch , 3Σ 3 2 ch M Ψ 2 = -3( r + ia cos θ ) 3 -3( r + ia cos θ ) 2 ( M + r + ia cos θ )( S 2 1 + S 2 2 ) --2 M ( r 2 +2 Mr -a 2 +3 iar cos θ + a 2 y 2 )( S 4 1 + S 4 2 )+ +12 M 2 S 2 1 S 2 2 ( M -r -ia cos θ )( S 2 1 + S 2 2 )+ +4 M S 2 1 S 2 2 (5 a 2 cos 2 θ -a 2 +4 Mr -5 r 2 +3 iaM cos θ -9 iar cos θ ) , Φ 00 = Φ 22 = M 2 ∆ r ( S 2 1 -S 2 2 ) 2 2Σ 3 2 ch , Φ 02 = Φ 20 = -a 2 M 2 sin 2 θ ( S 2 1 -S 2 2 ) 2 2Σ 3 2 ch , 3 2 ch M 2 Φ 11 = 4( r 2 + a 2 cos 2 θ )( S 2 1 + S 2 2 ) + (2 Mr +3 r 2 +3 a 2 cos 2 θ )( S 4 1 + S 4 2 )+ +16 M ( r + M ) S 2 1 S 2 2 ( S 2 1 + S 2 2 ) + 32 M 2 S 4 1 S 4 2 +2 S 2 1 S 2 2 ( r 2 +14 Mr + a 2 cos 2 θ ) , Λ = 1 24 R = -M 2 ( S 2 1 -S 2 2 ) 2 ( ∆ r + a 2 sin 2 θ ) 12Σ 3 2 ch . (119)  \nSimilar to the above cases of N = 4, the quantum separability condition (55) is also satisfied for the two-charge STU solution.  \nThe MPS equation reads  \nE 2 [ ∆ 2 ( r -M +2 M Σ s 2 ) -4 M 2 r ( Mr -a 2 )Σ 2 s 2 -16 M 4 ( r -M )Π 2 s 2 -8 M 3 ( r 2 -a 2 )Π s 2 Σ s 2 ] + +2 aELM [ 4 M ( r -M )Π s 2 -( r 2 -a 2 )Σ s 2 ] -a 2 L ( r -M ) -µ 2 ∆ 2 ( r -M + M Σ s 2 ) = 0 , (120)  \nwhere Σ s 2 = S 2 1 + S 2 2 , Π s 2 = S 2 1 S 2 2 .  \nFrom the above analysis of the various supergravity solutions for black holes, their similarity in the form of BF is obvious. For all of them, the coefficient functions A 1 , B 1 , A 23 , B 23 are just simple quadratic polynomials, which make us to believe that these solutions can be obtained by directly integrating the general supergravity equations.', 'VIII. CONCLUSIONS': "We have shown that the suitably refined Benenti-Francaviglia ansatz for stationary axisymmetric spacetimes, admitting a non-trivial Killing tensor of rank two, defines a class of non-algebraically special metrics that admits two null geodesic congruences without shear. If we additionally impose the separability condition of the Klein-Gordon equation, the resulting ansatz coincides with Carter's parametrization, which led him to perform a direct integration of  \nEinstein's vacuum and electrovacuum equations, reproducing, in particular, the Kerr-Newman metric and its extensions of type D. Our class of algebraically non-special metrics opens the way to a generalization of this approach to the general supergravity bosonic action (71) containing a set of vector and scalar fields. It was demonstrated explicitly that all known N = 2 , 4 , 8 supergravitational black holes, with the exception of the so-called four-charge solution, belong to our class indeed.  \nAn algebraically special subclass of our metrics contains only the type D . For this case, we have obtained an explicit parametrization of the BF coefficient functions integrating the Killing-Yano equations.  \nOur class of metrics leads to a general description of the confining surfaces of photons and massive particles that have recently been introduced in the theory of black hole shadows. The Killing tensor given by the BF formula has a crucial property of slice-reducibility [20] with respect to spacetime bundles by three-dimensional timelike hypersurfaces with constant r or y . This opens a way to universal description of supergravity black hole shadows. Here we have given a general equation for the confining surfaces of massive particles and its particular form for all examples of supergravity solutions.  \nTo summarize, the restricted Benenti-Francavilla metric gives a unified description of type I supergravity black holes, revealing many of their common properties. This was established by combining the Newman-Penrose analysis of the BF class of metrics with a specific form of known solutions. We hope that it will be possible to directly solve the supergravity equations of motion in the obtained parameterization, thus extending Carter's program to solve the Einstein equations for metrics admitting second-rank Killing tensors.", 'Acknowledgements': "The authors thanks Kirill Kobialko, Igor Bogush and G'erard Cl'ement for useful suggestions and discussions. This work was supported by Russian Science Foundation under Contract No. 23-22-00424.", 'Appendix A: Spin coefficients': 'Here we give NP spin coefficients calculated for the tetrad (17) without constraint (46) on the conformal factor Σ:  \nµ = ρ = -√ A 2 ∂ ∂ r 1 √ 2Σ -iB 5 √ B 2 2 √ 2 Σ ∂ y √ B 3 /B 5 √ A 3 B 5 -√ A 5 B 3 , (A1)  \n/epsilon1 = γ = 1 2 ρ + √ A 2 2 √ 2Σ 3 / 2 ( Σ ∂ r ln [ √ A 3 B 5 -√ A 5 B 3 ] -∂ r Σ ) , (A3)  \nτ = π = -i √ B 2 ∂ ∂ y 1 √ 2Σ + A 5 √ A 2 2 √ 2 Σ ∂ r √ A 3 /A 5 √ A 3 B 5 -√ A 5 B 3 , (A2)  \nα = β = 1 2 τ + i √ B 2 2 √ 2Σ 3 / 2 ( Σ ∂ y ln [ √ A 3 B 5 -√ A 5 B 3 ] -∂ y Σ ) . (A4)', 'Appendix B: Ricci and Weyl scalars': "With account for the second constraint the NP projections of te Ricci and Weyl scalars are  \nΦ 00 = Φ 22 = bA 2 8Σ 3 ( bA ' 23 2 + bB ' 23 2 -2Σ A '' 23 ) , (B1)  \nΦ 01 = Φ 10 = Φ 12 = Φ 21 = -√ A 2 B 2 8Σ 3 ( aA '' 23 -b B '' 23 ) , (B2)  \nΦ 02 = Φ 20 = -aB 2 8Σ 3 ( aA ' 23 2 + aB ' 23 2 +2Σ B '' 23 ) , (B3)  \n64Σ 3 Φ 11 = Σ 2 { A 2 [( ln A 5 ) ,r 2 -4 ( ln A 5 ) ,rr ] -B 2 [( ln B 5 ) ,r 2 -4 ( ln B 5 ) ,rr ] } + +3 { 4 b 2 A 2 B ' 23 2 -4 a 2 B 2 A ' 23 2 + A 2 [ Σ(ln A 5 ) ,r +2 bA ' 23 ] 2 -B 2 [ Σ(ln B 5 ) ,y -2 aB ' 23 ] 2 } + +4Σ { A 2 ((ln A 5 ) ,r ) [ Σ(ln A 5 ) ,r +2 bA ' 23 ] -A 2 [ Σ(ln A 5 ) ,r +2 bA ' 23 ] ,r } --4Σ { B 2 ((ln B 5 ) ,y ) [ Σ(ln B 5 ) ,y -2 aB ' 23 ] -B 2 [ Σ(ln B 5 ) ,y -2 aB ' 23 ] ,y } , (B4)  \n192 Σ 3 Λ = 8 Σ 3 R = 3Σ 2 { A 2 [ 3(ln A 5 ) ,r 2 -4(ln A 5 ) ,rr ] + B 2 [ 3(ln B 5 ) ,y 2 -4(ln B 5 ) ,yy ]} + +4 A 2 [ Σ(ln A 5 ) ,r +2 bA ' 23 ] ,r +4 B 2 [ Σ(ln B 5 ) ,y -2 aB ' 23 ] ,y -4 a 2 B 2 A ' 23 2 --A 2 [ Σ(ln A 5 ) ,r +2 bA ' 23 ] 2 -B 2 [ Σ(ln B 5 ) ,y -2 aB ' 23 ] 2 -4 b 2 A 2 B ' 23 2 . (B5)  \n-12Σ 3 Ψ 2 = Σ { A 2 [ Σ ( ln A 5 ) ,r +2 bA ' 23 ] ,r + B 2 [ Σ ( ln B 5 ) ,y -2 aB ' 23 ] ,y } +2 a 2 B 2 A ' 23 2 + -A 2 [ Σ ( ln A 5 ) ,r +2 bA ' 23 ] 2 -B 2 [ Σ ( ln B 5 ) ,y -2 aB ' 23 ] 2 +2 b 2 A 2 B ' 23 2 + +3 i { aB 2 A ' 23 [ Σ ( ln B 5 ) ,y -2 aB ' 23 ] + b A 2 B ' 23 [ Σ ( ln A 5 ) ,r +2 bA ' 23 ]} . (B6)  \n- [1] M. Cariglia, 'Hidden Symmetries of Dynamics in Classical and Quantum Physics,' Rev. Mod. Phys. 86 (2014), 1283 [arXiv:1411.1262 [math-ph]].\n- [2] V. P. Frolov, P. Krtous and D. Kubiznak, 'Black holes, hidden symmetries, and complete integrability,' Living Rev. Rel. 20 (2017) no.1, 6 [arXiv:1705.05482 [gr-qc]].\n- [3] U. Lindstrom and O. Sarıo˘glu, 'Uses of Killing-Yano Tensors,' PoS CORFU2021 (2022), 149 [arXiv:2202.07226 [hep-th]].\n- [4] Y. Chervonyi and O. Lunin, 'Killing(-Yano) Tensors in String Theory,' JHEP 09 (2015), 182 [arXiv:1505.06154 [hep-th]].\n- [5] P. Breitenlohner, D. Maison and G. W. Gibbons, 'Four-Dimensional Black Holes from KaluzaKlein Theories,' Commun. Math. Phys. 120 (1988), 295.\n- [6] P. Breitenlohner and D. Maison, 'On nonlinear sigma models arising in (super-)gravity,' Commun. Math. Phys. 209 (2000), 785-810 [arXiv:gr-qc/9806002 [gr-qc]].\n- [7] A. Bouchareb, G. Clement, C. M. Chen, D. V. Gal'tsov, N. G. Scherbluk and T. Wolf, 'G(2) generating technique for minimal D=5 supergravity and black rings,' Phys. Rev. D 76 (2007), 104032 [erratum: Phys. Rev. D 78 (2008), 029901] [arXiv:0708.2361 [hep-th]].\n- [8] D. V. Galtsov and O. V. Kechkin, 'Ehlers-Harrison type transformations in dilaton - axion gravity,' Phys. Rev. D 50 (1994), 7394-7399 [arXiv:hep-th/9407155 [hep-th]].\n- [9] E. Bergshoeff, R. Kallosh and T. Ortin, 'Stationary axion / dilaton solutions and supersymmetry,' Nucl. Phys. B 478 (1996), 156-180 [arXiv:hep-th/9605059 [hep-th]].\n- [10] D. Youm, 'Black holes and solitons in string theory,' Phys. Rept. 316 (1999), 1-232 [arXiv:hepth/9710046 [hep-th]].\n- [11] I. Bogush, G. Cl'ement, D. Gal'tsov and D. Torbunov, 'Nutty Kaluza-Klein dyons revisited,' Phys. Rev. D 103 (2021) no.6, 064045 [arXiv:2009.07922 [gr-qc]].\n- [12] R. R. Khuri and T. Ortin, 'Supersymmetric black holes in N=8 supergravity,' Nucl. Phys. B 467 (1996), 355-382 [arXiv:hep-th/9512177 [hep-th]].\n- [13] J. Bellorin and T. Ortin, 'All the supersymmetric configurations of N=4, d=4 supergravity,' Nucl. Phys. B 726 (2005), 171-209 [arXiv:hep-th/0506056 [hep-th]].\n- [14] G. Clement and D. V. Galtsov, 'Stationary BPS solutions to dilaton - axion gravity,' Phys. Rev. D 54 (1996), 6136-6152 [arXiv:hep-th/9607043 [hep-th]].\n- [15] G. Bossard, Y. Michel and B. Pioline, 'Extremal black holes, nilpotent orbits and the true fake  \nsuperpotential,' JHEP 01 (2010), 038 [arXiv:0908.1742 [hep-th]].  \n- [16] B. Carter, 'Hamilton-Jacobi and Schrodinger separable solutions of Einstein's equations,' Commun. Math. Phys. 10 (1968) no.4, 280-310.\n- [17] J. R. Gair and D. Lynden-Bell, 'Electromagnetic Fields of Separable Space-Times,' Class. Quant. Grav. 24 (2007), 1557-1567 [arXiv:gr-qc/0703003 [gr-qc]].\n- [18] S. Benenti and M. Francaviglia, 'Remarks on certain separability structures and their applications to general relativity,' Gen. Rel. Grav. 10 (1979) no.1, 79-92.\n- [19] K. Kobialko, I. Bogush and D. Gal'tsov, 'Killing tensors and photon surfaces in foliated spacetimes,' Phys. Rev. D 104 (2021) no.4, 044009 [arXiv:2104.02167 [gr-qc]].\n- [20] K. Kobialko, I. Bogush and D. Gal'tsov, 'Slice-reducible conformal Killing tensors, photon surfaces, and shadows,' Phys. Rev. D 106 (2022) no.2, 024006 [arXiv:2202.09126 [gr-qc]].\n- [21] K. Akiyama et al. [Event Horizon Telescope], 'First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,' Astrophys. J. Lett. 875 (2019), L1 [arXiv:1906.11238 [astro-ph.GA]].\n- [22] K. Akiyama et al. [Event Horizon Telescope], 'First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way,' Astrophys. J. Lett. 930 (2022) no.2, L12 [arXiv:2311.08680 [astro-ph.HE]].\n- [23] S. Benenti, C. R. Acad. Sci. Paric, serie A. Proc. Ser. A 283 , 215 (1976).\n- [24] S. Benenti, 'Separable dynamical systems: Characterization of separability structures on Riemannian manifolds,' Reports on Mathematical Physics 12 (1977), 311-316.\n- [25] J. A. Schouten, Kon. Ned. Akad. Wetensch. Proc. 43 , 449 (1940).\n- [26] J. A. Nijenhuis, Ned. Akad. Wetensch. Proc. Ser. A 58 , 390 (1955).\n- [27] M. Cariglia, P. Krtous and D. Kubiznak, 'Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets,' Phys. Rev. D 84 (2011), 024004 [arXiv:1102.4501 [hep-th]].\n- [28] G. O. Papadopoulos and K. D. Kokkotas, 'Preserving Kerr symmetries in deformed spacetimes,' Class. Quant. Grav. 35 (2018) no.18, 185014 [arXiv:1807.08594 [gr-qc]].\n- [29] G. O. Papadopoulos and K. D. Kokkotas, 'On Kerr black hole deformations admitting a Carter constant and an invariant criterion for the separability of the wave equation,' Gen. Rel. Grav. 53 (2021) no.2, 21 [arXiv:2007.12125 [gr-qc]].\n- [30] Z. Carson and K. Yagi, 'Asymptotically flat, parameterized black hole metric preserving Kerr  \nsymmetries,' Phys. Rev. D 101 (2020) no.8, 084030 [arXiv:2002.01028 [gr-qc]].  \n- [31] R. A. Konoplya, Z. Stuchl'ık and A. Zhidenko, 'Axisymmetric black holes allowing for separation of variables in the Klein-Gordon and Hamilton-Jacobi equations,' Phys. Rev. D 97 (2018) no.8, 084044 [arXiv:1801.07195 [gr-qc]].\n- [32] E. Newman and R. Penrose, 'An Approach to gravitational radiation by a method of spin coefficients,' J. Math. Phys. 3 (1962), 566-578.\n- [33] C. Keeler and F. Larsen, 'Separability of Black Holes in String Theory,' JHEP 10 (2012), 152 [arXiv:1207.5928 [hep-th]].\n- [34] J. Goldberg, R. Sachs, et al., 'A theorem on petrov types,' Acta Physica Polonica B, Proceedings Supplement 22 , 13 (1962).\n- [35] M. Cvetic, G. W. Gibbons and C. N. Pope, 'STU Black Holes and SgrA,' JCAP 08 (2017), 016 [arXiv:1705.05740 [gr-qc]].\n- [36] D. D. K. Chow and G. Comp'ere, 'Seed for general rotating non-extremal black holes of N = 8 supergravity,' Class. Quant. Grav. 31 (2014), 022001 [arXiv:1310.1925 [hep-th]].\n- [37] D. D. K. Chow and G. Comp'ere, 'Black holes in N=8 supergravity from SO(4,4) hidden symmetries,' Phys. Rev. D 90 (2014) no.2, 025029 [arXiv:1404.2602 [hep-th]].\n- [38] O. Y. Tsupko and G. S. Bisnovatyi-Kogan, 'Gravitational lensing in plasma: Relativistic images at homogeneous plasma,' Phys. Rev. D 87 (2013) no.12, 124009 [arXiv:1305.7032 [astro-ph.CO]].\n- [39] J. Bad'ıa and E. F. Eiroa, 'Shadow of axisymmetric, stationary, and asymptotically flat black holes in the presence of plasma,' Phys. Rev. D 104 (2021) no.8, 084055 [arXiv:2106.07601 [gr-qc]].\n- [40] B. Bezdekova, V. Perlick and J. Bicak, 'Light propagation in a plasma on an axially symmetric and stationary spacetime: Separability of the Hamilton-Jacobi equation and shadow,' J. Math. Phys. 63 (2022) no.9, 092501 [arXiv:2204.05593 [gr-qc]].\n- [41] V. Perlick and O. Y. Tsupko, 'Light propagation in a plasma on Kerr spacetime. II. Plasma imprint on photon orbits,' Phys. Rev. D 109 (2024) no.6, 064063 [arXiv:2311.10615 [gr-qc]].\n- [42] B. Carter, 'Killing Tensor Quantum Numbers and Conserved Currents in Curved Space,' Phys. Rev. D 16 (1977), 3395-3414.\n- [43] I. Bogush, G. Cl'ement and D. Gal'tsov, 'Mass formulas for supergravity black holes with string singularities,' Eur. Phys. J. C 84 (2024) no.7, 727 [arXiv:2405.19196 [gr-qc]].\n- [44] C. M. Claudel, K. S. Virbhadra and G. F. R. Ellis, 'The Geometry of photon surfaces,' J. Math. Phys. 42 (2001), 818-838 [arXiv:gr-qc/0005050 [gr-qc]].  \n- [45] K. S. Virbhadra and G. F. R. Ellis, 'Gravitational lensing by naked singularities,' Phys. Rev. D 65 (2002), 103004.\n- [46] G. W. Gibbons and C. M. Warnick, 'Aspherical Photon and Anti-Photon Surfaces,' Phys. Lett. B 763 (2016), 169-173 [arXiv:1609.01673 [gr-qc]].\n- [47] E. Teo, 'Spherical orbits around a Kerr black hole,' Gen. Rel. Grav. 53 (2021) no.1, 10 [arXiv:2007.04022 [gr-qc]].\n- [48] K. V. Kobialko and D. V. Gal'tsov, 'Photon regions and umbilic conditions in stationary axisymmetric spacetimes: Photon Regions,' Eur. Phys. J. C 80 (2020) no.6, 527 [arXiv:2002.04280 [gr-qc]].\n- [49] I. Bogush, K. Kobialko and D. Gal'tsov, 'Massive particle surfaces,' Phys. Rev. D 108 (2023) no.4, 044070.\n- [50] K. Kobialko, I. Bogush, and D. Gal'tsov, 'Black hole shadows of massive particles and photons in plasma,' (2023).\n- [51] G. Pappas and K. Glampedakis, 'On the connection of spacetime separability and spherical photon orbits,' (2018).\n- [52] K. Glampedakis and G. Pappas, 'Modification of photon trapping orbits as a diagnostic of nonKerr spacetimes,' Phys. Rev. D 99 (2019) no.12, 124041 [arXiv:1806.09333 [gr-qc]].\n- [53] R. A. Konoplya and A. Zhidenko, 'Shadows of parametrized axially symmetric black holes allowing for separation of variables,' Phys. Rev. D 103 (2021) no.10, 104033 [arXiv:2103.03855 [gr-qc]].\n- [54] A. Sen, 'Rotating charged black hole solution in heterotic string theory,' Phys. Rev. Lett. 69 (1992), 1006-1009 [arXiv:hep-th/9204046 [hep-th]]."}