Verbally closed subgroups of free solvable groups

We establish a series of results on verbally closed and $l$-verbally closed subgroups of free solvable groups; here $l$ is a natural number and the concept of $l$-verbal closedness is a generalization of the concept of verbal closedness corresponding to the value $l=1$. Under certain assumptions, these subgroups turn out to be retracts and, consequently, are algebraically closed.