Dynamics in the transition case invertible/non-invertible in a 2D Piecewise Linear Map

•We investigate the dynamics of a two-dimensional piecewise linear map at the transition invertible/noninvertible.•We show how complex the dynamics may be also in such degenerate cases (with multistability and chaotic dynamics).•We show how the use of the first return map in a suitable segment may help in some cases.•We prove a regime in which a 3-cycle attracts almost all the points: the whole plane except for three curves.