  The study of periodic solutions with constant sign in the Abel equation of
the second kind can be made through the equation of the first kind. This is
because the situation is equivalent under the transformation $x\mapsto x^{-1}$,
and there are many results available in the literature for the first kind
equation. However, the equivalence breaks down when one seeks for solutions
with nonconstant sign. This note is devoted to periodic solutions with
nonconstant sign in Abel equations of the second kind. Specifically, we obtain
sufficient conditions to ensure the existence of a periodic solution that
shares the zeros of the leading coefficient of the Abel equation. Uniqueness
and stability features of such solutions are also studied.
