  In this work, we present sufficient conditions for the existence of a
stationary solution of an abstract stochastic Cauchy problem driven by an
arbitrary cylindrical L\'evy process, and show that these conditions are also
necessary if the semigroup is stable, in which case the invariant measure is
unique. For typical situations such as the heat equation, we significantly
simplify these conditions without assuming any further restrictions on the
driving cylindrical L\'evy process and demonstrate their application in some
examples.
