On equations in free groups with commutant restrictions on solutions

A polynomial algorithm has been constructed that allows, given an arbitrary equation of the form $w(x_{1},\ldots,x_{n})=[a,b]$, resolved with respect to unknowns, where $w(x_{1},\ldots,x_{n})$ is a group word in the alphabet of unknowns and $[a,b]$ is the commutator of free generators $a$ and $b$ of the free group $F_2$, to determine whether there is a solution to this equation that satisfies the condition $x_{1}\ldots, x_{n}\in F_{2}^{(1)}$, where $F_2^{(1)}$ is the commutator of group $F_2$. The existence of a polynomial algorithm has been established that allows, given an arbitrary equation of the form $ w (x_{1}, \ldots, x_{n}) = g (a, b) $, where $ g (a, b) $ is an element of length less than $4$ of the free group $ F_{2} $, to determine whether a solution to this equation exists, that satisfies the condition $x_{1},\ldots, x_{t}\in F_{2}^{(1)}$, where $t$ is an arbitrary fixed number between 1 and $n$. The algorithmic solvability of a similar problem has been proven for the equations $w(x_{1},a,b)=1$ with one variable $x_1$.