Invariant subspaces of a generalized backward shift operator and rational functions

We obtaine the full characterization of proper closed invariant subspaces of a generalized backward shift operator (Pommiez operator) in the Frechet space of all holomorphic functions on a simply connected domain $\Omega$ of the complex plane, containing the origin. In the case when the function, which defines this operator, is not has zeros in $\Omega$ all such subspaces are finite-dimensional. If additionally $\Omega$ coincides with the complex plane, then the considered operator is unicellular. If this function has zeros in $\Omega$, then the family of mentioned invariant subspaces splits into two classes: the first consists of finite-dimensional subspaces, and the second of infinite-dimensional ones.