  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.
