Geodesic
Finite parametrization of solutions of equations in a free
Sbornik. Mathematics, Tome 195 (2004) no. 2, pp. 187-235

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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. 
@article{SM_2004_195_2_a2,
     author = {G. S. Makanin},
     title = {Finite parametrization of solutions of equations in a free},
     journal = {Sbornik. Mathematics},
     pages = {187--235},
     publisher = {mathdoc},
     volume = {195},
     number = {2},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2004_195_2_a2/}
}
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G. S. Makanin. Finite parametrization of solutions of equations in a free. Sbornik. Mathematics, Tome 195 (2004) no. 2, pp. 187-235. http://geodesic.mathdoc.fr/item/SM_2004_195_2_a2/