  By analytic perturbations, we refer to shifts that are finite rank
perturbations of the form $M_z + F$, where $M_z$ is the unilateral shift and
$F$ is a finite rank operator on the Hardy space over the open unit disc. Here
shift refers to the multiplication operator $M_z$ on some analytic reproducing
kernel Hilbert space. In this paper, we first isolate a natural class of finite
rank operators for which the corresponding perturbations are analytic, and then
we present a complete classification of invariant subspaces of those analytic
perturbations. We also exhibit some instructive examples and point out several
distinctive properties (like cyclicity, essential normality, hyponormality,
etc.) of analytic perturbations.
