Exact solutions of Maxwell equations in homogeneous spaces with the group of motions $G_3(VIII)$

The problem of classification of exact solutions of Maxwell's vacuum equations for admissible electromagnetic fields and homogeneous space-time with the group of motions $G_3(VIII)$ according to the Bianchi classification is considered. All non-equivalent solutions are found. The classification problem for remaining groups of motions $G_3(N)$ has already been solved in the other papers. That is why all non-equivalent solutions of empty Maxwell equations for all homogeneous spaces with admissible electromagnetic fields are known now.


Introduction
If the symmetry of space-time and physical fields is given by Killing fields whose number is not less than three, it is possible to reduce the field equations and the equations of motion of the tested charged particles to the systems of ordinary differential equations.
The spaces admitting complete sets of mutually commutative Killing tensor fields of rank not greater than two are of special interest in the theory of gravitation. Such spaces are called Steckel spaces. The theory of Steckel spaces was developed in [1], [2], [3], [4], [5], [6], [7]. (see also [8], [9] [10], [11], and the bibliography is given there). The equations of motion of test particles in Stackel spaces can be integrated using the commutative integration method -CIM (or the method of complete separation of variables). Exact solutions of the gravitational equations are still actively used in the study of various aspects of gravitational theory and cosmology (see, for example, [12], [13], [14], [15], [17], [18], [19], [20], [21], [22], [23], [24]. Another method for exact integration of the equations of motion for a test particle (the method of non-commutative integration (NCIM)) was proposed in [25]. The method is applied to spaces admitting non-commutative groups of motion G r (r), r ≥ 3 (see A. Petrov [26]). It allows for reducing the equations of motion to systems of ordinary differential equations. By analogy with Stackel spaces, we call them poststack spaces -PSS. PSS are also actively studied in gravitational theory and cosmology (see, e.g., [27], [28], [29], [31], [32], [33], [34], [35]). The classification of electromagnetic fields in which the Klein-Gordon-Fock equations and Hamilton-Jacobi equations admit non-commutative algebras of symmetry operators for a charged sample particle is carried out in [36], [37], [38], [39], The commutative and non-commutative integration methods have a similar classification problem, namely enumerating all non-equivalent metrics and electromagnetic potentials satis-fying the requirements of the given symmetry. For Stackel spaces, the problem of classifying admissible external electromagnetic fields and electrovacuum solutions of the Einstein-Maxwell equations was solved in [16].
In the previous works ( [40], [41], [42]), the non-null PSS of all types were considered according to the Bianchi classification except the type V III. In the present work, all non-equivalent exact solutions of Maxwell's vacuum equations for non-null PSS of type V III are obtained. Thus this classification is completed for all non-null PSS.

Admissible electromagnetic fields in homogeneous spaces
According to the definition (see [43]) the space-time V 4 is homogeneous if its metric can be represented in a semi-geodesic coordinate system as follows: is satisfied. Here e α a are the triad of the dual vectors: C a bc are structural constants of the group G 3 (N), which acts on V 4 . The vectors of the frame e a α define a non-holonomic coordinate system in the hypersurface of transitivity V 3 of the group G 3 (N). Here and elsewhere, dots denote the derivatives on the variable u 0 . The coordinate indices of the semi-geodesic coordinate system are denoted by letters: i, j, k = 0, 1 . . . 3. The variables of the local coordinate system on V 3 are provided with indices: α, β, γ = 1, . . . 3. Indices of non-holonomic frame are provided with indices: a, b, c = 1, . . . 3. The rule is used, according to which the repeating upper and lower indices are summarized within the index range.
It has been proved in the paper [37], that for a charged test particle moving in the external electromagnetic field with potential A i , the Hamilton-Jacobi equation: and the Klein-Gordon-Fock equation: admit the integrals of motion if and only if the conditions: are satisfied. Here p i = ∂ i ϕ;p k = −ı∇ k ; (∇ k is the covariant derivative operator corresponding to the partial derivative operator -∂ i , ϕ is a scalar function of the particle with mass m); ξ i α is Killing vectors, C c ab are structural constants: If A i satisfies condition (7), the electromagnetic field is called admissible. All admissible electromagnetic fields for groups of motion G r (N) (r ≥ 3) acting transitively on hypersurfaces of the space-time have been found in [37], [38], [39].
Let us show that solutions of the system of equations (7) for HPSS of type V III can be represented in the form: To prove this, let us find the frame vector using the metric tensor of Bianchi's V III-type space (see [26]).
To obtain the functions e α a it is sufficient to consider the components g 11 , g 12 , g 13 from the system (1). The solution can be represented in the form: The lower index numbers the lines. The solution of the system of equations (7) has been found in [37]. It has the form:

Maxwell's equations
All exact solutions of empty Maxwell's equations for solvable groups have been found in the papers [40], [41]. The present paper solves the problem for the group G 3 (V III).
Consider empty Maxwell's equations for an admissible electromagnetic field in homogeneous space with a group of motions G r : The metric tensor and the electromagnetic potential are defined by relations (1), (8). When i = 0, from the set of equations (11) it follows: Here it is denoted: Let i = α. Then, from equation (11), it follows: Let us find components of F αβ using the relations (8).
We present the structural constants of a group G 3 in the form: where Equations (16) will take the form: To decrease the order of the equations (18), we introduce new independent functions: Let us introduce the function: Then Maxwell's equations (18) and (21) take the form of a system of linear algebraic equations on the unknown functions n ab : The equation (19): is a restriction on the function b a (if ρ a = 0). Let us obtain the Maxwell's equations for the group G 3 (V III). Non-zero structural constants, in this case, have the form: From here, it follows: Using these relations, we obtain Maxwell's equations (18) -in the form: n T = (n 11 , n 12 , n 13 , n 22 , n 23 , n 33 );ω T = (−ḃ 2 ,ȧ 2 , −ḃ 1 ,ȧ 1 ,ḃ ).
Here and after next notations are used: Let us find the algebraic complement of the matrixB : AsB is a singular matrix,V is the annulling matrix forB: So when v 1 2 + v 2 2 + v 3 2 = 0, one of the equations from the system (26) can be replaced by the equation: Depending on the rank of the matrixB, one or more functions n ab (u 0 ) are independent. It is possible to express the remaining functions n ab through the functions a a , b a . To find non-equivalent solutions of the system (26), one should consider the following variants: 1. a 1 = 0; 2. a 1 = 0, a 2 = 0; 3. a 1 = a 2 = 0, a 3 = 0. Taking this observation into account, let us consider all non-equivalent options.

Solutions of Maxwell equations
Since the functions a a satisfy the condition: rank of matrix (29) cannot be less than 3. If In order to obtain a complete solution to the classification problem, it is necessary: I to consider all non-equivalent variants with non-zero minors of rank = 5 of the matrixB; II to consider all non-equivalent variants under the condition: v a = 0 (rank ≤ 3).
The components of the matrixη as well as the functions α a are given by formulae (21), (28) In view of these circumstances, let us list all exact solutions of empty Maxwell equations for PSS of type VIII. I. rank||B|| = 5. 1. a 1 v 1 = 0 ⇒ the minorB 12 and its inverse matrixP =B −1 12 have the form: Then the solution of equation (26) is as follows: weren T 1 = (n 12 , n 13 , n 22 , n 23 , n 33 ); Function n 11 , a a , b a are arbitrary functions of u 0 , that obey the condition (31).
2. a 2 v 1 = 0. Obviously, we obtain a non-equivalent solution to the previous one only if a 1 = 0. In order to implement the classification, a similar choice should be made for all other variants. The matrixB 14 and its inverse matrixP 2 =B −1 14 have the form: Then the solution of equation (26) is as follows: weren T 2 = (n 12 , n 13 , n 22 , n 23 , n 33 ); Function n 11 , a a , β a are arbitrary functions of u 0 , that obey the condition (31).

Conclusion
In the previous works [ [40] [41], [42] ] ], all non-equivalent solutions of Maxwell's empty equations for admissible electromagnetic fields in homogeneous space-time metrics of all types according to Bianchi's classification, except type V III, were found. The present work completes the first stage of the classification problem formulated in the introduction. The next step is the classification of the corresponding exact solutions of the Einstein-Maxwell equations. All solutions obtained in the completed classification have a form suitable for further use and have sufficient arbitrariness so that the Einstein-Maxwell equations have nontrivial solutions. The use of the triad of frame vectors (see [43]) allows us to reduce the Einstein-Maxwell equations with the energy-momentum tensor of the admissible electromagnetic field to an overcrowded system of ordinary differential equations. To perform the classification, we need to study the coexistence conditions of these systems of equations. It is possible to use additional symmetries of homogeneous spaces and admissible electromagnetic fields (see [39]). In the future, we will start to solve this classification problem.