  We discuss a technique to construct Ricci-flat hermitian metrics on
complements of (some) anticanonical divisors of almost homogeneous manifolds
and discuss when this metric is complete and K\"ahler. This construction has a
strong interplay with invariance groups of the same dimension as the manifold
acting with an open orbit. Lie groups of this type we call divisorial. As an
application we can describe compact manifolds admitting a divisorially
invariant K\"ahler metric on an open subset. Finally, we see a connection
between the reducibility of the anticanonical divisor and the non-triviality of
the K\"ahler cone on the complement.
