On the representation of substitutions as products of a transposition and a full cycle

A method of solving equations of the form $g^{y_1}\cdot h\cdot g^{y_2}\cdot h\cdot\ldots\cdot g^{y_l}\cdot h\cdot g^{y_{l+1}}=\sigma$ in the symmetric group $\mathrm S_n$ is proposed, where $h$ is a transposition, $g$ is a full cycle, and $\sigma\in\mathrm S_n$. The method is based on building all sets of generalized inversions of the bottom line of the substitution $\sigma$ by means of a system of Boolean equations associated with $\sigma$. An example of solving an equation in a group $\mathrm S_6$ is given.