Parametrization of the Solutions of the Equation $x_1x_2\dots x_{n-1}x_n=x_nx_{n-1}\dots x_2x_1$ in a Free Monoid

A parametrizing function $\mathrm{Sm}$ is introduced. The parametrizing function is a recursive function depending on lexicographic variables, natural variables, and variables whose values are finite sequences of natural variables. Using the function $\mathrm{Sm}$, we construct formulas that provide all the solutions of the equation
$$ x_1x_2\dots x_{n-1}x_n=x_nx_{n-1}\dots x_2x_1 $$
in a free monoid $\langle a_1,a_2,\dots,a_\omega\rangle$ and only them.