  The complexity of the equation solvability problem is known for nilpotent
groups, for not solvable groups and for some semidirect products of Abelian
groups. We provide a new polynomial time algorithm for deciding the equation
solvability problem over certain semidirect products, where the first factor is
not necessarily Abelian. Our main idea is to represent such groups as matrix
groups, and reduce the original problem to equation solvability over the
underlying field. Further, we apply this new method to give a much more
efficient algorithm for equation solvability over nilpotent rings than
previously existed.
