Group reduction of the Lamé equations (Q913001)

A system of Lamé equations describes the equilibrium state of an n- dimensional homogeneous elastic medium in the absence of mass forces. The transformations of the space \({\mathcal R}^{2n}\) admissible by the system in question, form an infinite normal subgroup over which the factor-group becomes finite-dimensional. The action of this transformation group on solutions of the given system of Lamé equations induces their fine structure describable in terms of two systems of first-order differential equations refered to as the automorphic and solvable classes. The general solution of the former class is a multidimensional analogue of the Kolosov-Muskhelishvili formula. The latter system becomes conformally- invariant in \({\mathcal R}^ 3\). That is why it possesses Kelvin-like transformations whose general explicit form is presented.