Affine pseudo-coverings of algebraic surfaces

An affine pseudo-covering f:Y→X of smooth affine varieties is an étale morphism whose image contains all codimension one points of X. If f splits an étale endomorphism φ:X→X as φ=f⋅g with a dominant morphism g:X→Y, then f and g are affine pseudo-coverings under some additional conditions which are satisfied when X is the affine n-space An. Motivated by the Jacobian problem, we consider an affine pseudo-coverings in the case where Y or X is isomorphic to the affine plane A2.