Extensions of Lie algebras and integrability of some equations of fluid dynamics and magnetohydrodynamics.

We find the twisted extension of the symmetry algebra of the 2D Euler equation in the vorticity form and use it to construct new Lax representation for this equation. Then we consider the transformation Lie-Rinehart algebras generated by finite-dimensional subalgebras of the symmetry algebra and employ them to derive a family of Lax representations for the Euler equation. The family depends on functional parameters and contains a non-removable spectral parameter. Furthermore we exhibit Lax representations for the reduced magnetohydrodynamics equations (or the Kadomtsev-Pogutse equations), the ideal magnetohydrodynamics equations, the quasigeostrophic two-layer model equations, and the Charney-Obukhov equation for the ocean.