We generalize the notion of a distributive law between a monad and
a comonad to consider weakened structures such as pointed and 
co-pointed endofunctors, or endofunctors. 
We investigate Eilenberg-Moore and Kleisli constructions for each
of these possibilities. Then we consider two applications of these
weakened notions of distributivity in detail. 
We characterize Turi and Plotkin's model of GSOS as a distributive 
law of a monad over a co-pointed endofunctor, and we anlyse generalized
coiteration and coalgebraic coinduction "up-to" in terms of a 
distributive law of the underlying pointed endofunctor of a monad
over an endofunctor.