left eigenvector as unique solution?

Question

Consider the following equation in linear algebra: $\ \lambda·x·y-(x·M·y)=0$ , with $\ x$ and $\ y$ being respectively 1xn and nx1 vectors, and $\ M$ a nxn real semipositive and irreducible matrix. Also vector $\ y$ is real semipositive (but not one of the right eigenvectors of $\ M$), while $\lambda$ is the dominant eigenvalue of $\ M$. My aim is to find vector $\ x$. Certainly, from what I can see, apart the trivial one, a solution is the dominant left eigenvector of $\ M$. The question I ask if it is the only non trivial solution or if other solutions are also possibile.