  We extend the so-called retract relation given in [6] for involutive
set-theoretic solutions of the Pentagon Equation and we introduce the notion of
associated permutation group to study the family of the commutative
non-degenerate ones. Moreover, we develop a machinery to construct all these
solutions and we use it to give a quite explicit classification of the
irretractable ones. Finally, non-degenerate solutions on left-zero semigroup
are studied in detail, with an emphasis on the ones with cyclic associated
permutation group and on the ones having small size.
