Integrable and non-integrable structures in Einstein - Maxwell equations with Abelian isometry group $\mathcal{G}_2$

Abstract

The classes of electrovacuum Einstein - Maxwell fields (with a cosmological constant), which metrics admit an Abelian two-dimensional isometry group $\mathcal{G}_2$ with non-null orbits and electromagnetic fields possess the same symmetry, are considered. For these fields we describe the structures of so called "nondynamical degrees of freedom" which presence as well as the presence of a cosmological constant change (in a strikingly similar ways) the dynamical equations and destroy their known integrable structures. Modifications of the known reduced forms of Einstein - Maxwell equations -- the Ernst equations and self-dual Kinnersley equations in the presence of non-dynamical degrees of freedom are found and subclasses of fields with non-dynamical degrees of freedom are considered for : (I) vacuum metrics with cosmological constant, (II) vacuum space-times with isometry groups $\mathcal{G}_2$ which orbits do not admit the orthogonal 2-surfaces (none-orthogonally-transitive isometry groups) and (III) electrovacuum fields with more general structures of electromagnetic fields than in the known integrable cases. For these classes, in the case of diagonal metrics, the field equations can be reduced to the only nonlinear equation of the fourth order for one real function $\alpha(x^1,x^2)$ which characterises the element of area on the orbits of the isometry group $\mathcal{G}_2$ . Simple examples of solutions for each of these classes are presented. It is pointed out that if for some two-dimensional reduction of Einstein's field equations in four or higher dimensions, the function $\alpha(x^1,x^2)$ possess a "harmonic" structure, instead of being (together with other field variables) a solution of some nonlinear equations, this can be an indication of possible complete integrability of these reduced dynamical equations for the fields with vanishing of all non-dynamical degrees of freedom.