  For a given quadratic equation with any number of unknowns in any free group
F, with right-hand side an arbitrary element of F, an algorithm for solving the
problem of the existence of a solution was given by Culler. The problem has
been studied by the authors for parametric families of quadratic equations
arising from continuous maps between closed surfaces, with certain conjugation
factors as the parameters running through the group F. In particular, for a
one-parameter family of quadratic equations in the free group F_2 of rank 2,
corresponding to maps of absolute degree 2 between closed surfaces of Euler
characteristic 0, the problem of the existence of faithful solutions has been
solved in terms of the value of the self-intersection index mu: F_2 --> Z[F_2]
on the conjugation parameter. The present paper investigates the existence of
faithful, or non-faithful, solutions of similar families of quadratic equations
corresponding to maps of absolute degree 0.
