  It is shown that square free set theoretic involutive non-degenerate
solutions of the Yang-Baxter equation whose associated permutation group
(referred to as an involutive Yang-Baxter group) is abelian are retractable in
the sense of Etingof, Schedler and Soloviev. This solves a problem of
Gateva-Ivanova in the case of abelian IYB groups. It also implies that the
corresponding finitely presented abelian-by-finite groups (called the structure
groups) are poly-${\mathbb Z}$ groups. Secondly, an example of a solution with
an abelian involutive Yang-Baxter group which is not a generalized twisted
union is constructed. This answers in the negative another problem of
Gateva-Ivanova. The constructed solution is of multipermutation level 3.
Retractability of solutions is also proved in the case where the natural
generators of the IYB group are cyclic permutations. Moreover, it is shown that
such solutions are generalized twisted unions.
