I will start with a review of some physical phenomena and applications involving plasmas, and present the corresponding model PDEs. We will concentrate on the continuum isotropic model of plasmas – the system of Magnetohydrodynamics (MHD) equations. It is a fully nonlinear 3D system. Its equilibrium version is known for its very special solution topology. I will discuss necessary physical properties of equilibrium solutions, overview the known methods of construction of exact solutions, and present several families of exact solutions with relevant physical models.

One of possible ways of construction of new exact solutions to a PDE system is symmetries – maps of the solution manifold into itself. I will show how the rich symmetry structure of equilibrium MHD equations significantly extends and diversifies the set of known exact solutions (and corresponding physical models).

Differences, similarities and relations of MHD with Navier-Stokes and Euler models will be outlined.