  In this article, the existence of a unique solution in the variational
approach of the stochastic evolution equation $$\dX(t) = F(X(t)) \dt + G(X(t))
\dL(t)$$ driven by a cylindrical L\'evy process $L$ is established. The
coefficients $F$ and $G$ are assumed to satisfy the usual monotonicity and
coercivity conditions. The noise is modelled by a cylindrical L\'evy processes
which is assumed to belong to a certain subclass of cylindrical L\'evy
processes and may not have finite moments.
