An application of Lax’s formalism to a class of curvilinear axially symmetric coordinates in three-dimensional space

Abstract

Solutions of a nonlinear system of differential partial equations in two-dimensional space are studied. We prove that the considered system is fully integrable and that its solutions determine an infinite class of mappings of the space of curvilinear axially symmetric coordinates onto the space of cylindric coordinates. The corresponding family of reverse transformations is found in a closed form. An infinite number of the first integrals of the nonlinear system is obtained. Symmetries of a class of electrostatic systems associated with the considered family of curvilinear coordinates are discussed.