  We probe deeper into the Discrete Markus-Yamabe Question for polynomial
planar maps and into the normal form for those maps which answer this question
in the affirmative. Furthermore, in a symmetric context, we show that the only
nonlinear equivariant polynomial maps providing an affirmative answer to the
Discrete Markus-Yamabe Question are those possessing Z2 as their group of
symmetries. We use this to establish two new tools which give information about
the spectrum of a planar polynomial map.
