On recursion operators for symmetries of the Pavlov-Mikhalev equation

Abstract: In geometry of nonlinear partial differential equations, recursion operators that act on symmetries of an equation $\mathcal{E}$ are understood as B\"{a}cklund auto-transformations of the equation $\mathcal{TE}$ tangent to $\mathcal{E}$. We apply this approach to a natural two-component extension of the 3D Pavlov-Mikhalev equation \begin{equation*} u_{yy} = u_{tx} + u_yu_{xx} - u_xu_{xy}. \end{equation*} We describe the Lie algebra of symmetries for this extension, construct two recursion operators (one of them was known earlier) and find their action. We also establish the hereditary property of these operators as well as their compatibility (in the sense of the Fr\"{o}licher-Nijenhuis bracket). We find also twelve additional operators which are degenerate in a sense (we call them \emph{queer}) and discuss their properties. In the concluding part, a geometrical background of two-component conservation laws for multi-dimensional equations is exposed together with its relations to differential coverings.