First Integrals and Exact Solutions of the Generalized Models of Magnetic Insulation

We suggest generalizations of the mathematical model of magnetic insulation, described by multidimensional quasi potential ODE system or PDE system with two-dimensional Laplace operator. Existence conditions of the first integrals of a certain type for the class of nonlinear quasi potential systems, including the model vacuum diode are obtained. Integrability of the vacuum diode models is justified. We find for PDE system the class of exact radially symmetric solutions given by fractional-rational functions. The class of systems with variable density, reduced to a similar system with the constant current density by special transformations is specified. The class of exact solutions of the non-singular boundary-value problem in annular domain is found.