Parametrization of solutions of some equations of squares in a free monoid

We introduce parametrizing functions $Fa$, $La$, $Do$ of word variables, natural number variables, and variables whose values are finite sequences of natural number variables. By means of parametrizing functions $Fa$, $La$, $Do$ we give finite formulas for the family of solutions of the equation
$$ x_1^2x_2^2x_3^2=x_4^2 $$
in a free monoid.
The research was supported by the Russian Foundation for Basic Research, grant 98–01–00885.