Geodesic
Finite parametrization of solutions of~equations in a~free
Sbornik. Mathematics, Tome 195 (2004) no. 4, pp. 521-552

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In the preceding paper of the author parametrizing functions Fi, Th, Ro were introduced depending on word variables, positive-integer variables, and variables whose values are finite sequences of positive-integer variables. With the help of the parametrizing functions Fi, Th, Ro finite formulae are written out for the family of solutions of every equation of the form $\varphi(x_1,x_2,x_3) x_4=\psi(x_1,x_2,x_3) x_5$, where $\varphi(x_1,x_2,x_3)$ and $\psi(x_1,x_2,x_3)$ are arbitrary words in the alphabet $x_1$, $x_2$, $x_3$ in a free monoid. 
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     author = {G. S. Makanin},
     title = {Finite parametrization of solutions of~equations in a~free},
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G. S. Makanin. Finite parametrization of solutions of~equations in a~free. Sbornik. Mathematics, Tome 195 (2004) no. 4, pp. 521-552. http://geodesic.mathdoc.fr/item/SM_2004_195_4_a2/