Variations on the theme of Zariski's Cancellation Problem

I call an algebraic variety flattenable if it is isomorphic to an open subset of an affine space. In this talk, I discuss, in the frame of algebraic group actions, the topic whose starting point is the following question that resembles the Zariski Cancellation Problem: Are there affine algebraic varieties X and Y such that X and X\times Y are flattenable, but Y is not?