Parametrization of solutions of the equation $x^{-1}y^{-1}xyz^{-1}v^{-1}zv=1$ in a free group

A finite parametrisation of the set of all solutions of the equation
$$ x^{-1}y^{-1}xyz^{-1}v^{-1}zv=1 $$
in a finite group is given. This equation is closely related to the Poincare conjecture. The parametrising functions depend on word variables, natural variables, and variables, whose values are sequences of natural variables.
This research was supported by the Russian Foundation of Basic Research, grant 01–01–00027.