  A necessary and sufficient condition is given for general exact remote state
preparation (RSP) protocols to be oblivious, that is, no information about the
target state can be retrieved from the classical message. A novel criterion in
terms of commutation relations is also derived for the existence of
deterministic exact protocols in which Alice's measurement eigenstates are
related to each other by fixed linear operators similar to Bob's unitaries. For
non-maximally entangled resources, it provides an easy way to search for RSP
protocols. As an example, we show how to reduce the case of partially entangled
resources to that of maximally entangled ones, and we construct RSP protocols
exploiting the structure of the irreducible representations of Abelian groups.
