New bad lines in R[x,y] and optimization of the Epimorphism Theorem

We give necessary and sufficient conditions on a commutative ring with unity R so that the following holds: any polynomial f in R[x,y] such that R[x,y]/(f) R[1] (a line for short) is a variable (=α(x) for some automorphism ). The main ingredient here is the construction of new bad lines (lines which are not variables).