  This paper develops an analytic theory for the study of some Polya urns with
random rules. The idea is to extend the isomorphism theorem in Flajolet et al.
(2006), which connects deterministic balanced urns to a differential system for
the generating function. The methodology is based upon adaptation of operators
and use of a weighted probability generating function. Systems of differential
equations are developed, and when they can be solved, they lead to
characterization of the exact distributions underlying the urn evolution. We
give a few illustrative examples.
