On the structure polynomial of first order linear systems on the plane

We consider the hydrodynamics and magnethydrodynamic models from point of view the Vekua's theory of Generalized Analytic Functions. Also, the discrete case has been recently studied. An important result of Vekua's theory of generalized analytic functions is the construction of the canonical form for uniformly and linear elliptic system of equations on the plane by solving an associated Beltrami equation. In this talk we will introduce the concept of structure polynomial of linear first order systems on the plane and show how the induced algebraic structure allows to avoid solving a Beltrami equation for a family of cases even when the system is not uniformly elliptic.