One variable equations over the lamplighter group

We prove that one variable equations in the lamplighter group ${\text{MZ}}_2\wr \text{MZ}$ are decidable and describe an algorithm for solving such equations. The algorithm has super-exponential time complexity in the worst case. We also show that, for most equations, decidability can be determined in nearly quadratic time; that is, the problem admits a nearly quadratic-time solution in the generic case.