Finite parametrization of solutions of equations in a free monoid. I

Complete solutions of all primitively parametrizable basis equations in a free monoid are written out. Parametrizing functions Fi, Th, Ro are introduced depending on word variables, positive integer variables, and variables whose values are finite sequences of positive-integer variables. Finite formulae for the family of solutions of all Fi-, Th-, Ro-parametrizable basis equations of the form $x_1x_2x_3x_4=\psi(x_1,x_2,x_3)$ in a free monoid are written out with the help of the parametrizing functions.