Sets of polynomial rings in two variables and factorizations of polynomial morphisms

Let k be a field. We study infinite strictly descending sequences A0⊃A1⊃⋯A0⊃A1⊃⋯ of rings where each AiAi is a polynomial ring in two variables over k, the aim being to describe those sequences satisfying ⋂i=0∞Ai≠k. We give a complete answer in characteristic zero, and partial results in arbitrary characteristic. We apply those results to the study of dominant morphisms A2→AnA2→An and their factorizations, where n∈{1,2}n∈{1,2} and AnAn is the affine n-space over k.