  We introduce the notion of multiplication kernels of birational and
$D$-module type and give various examples. We also introduce the notion of a
semi-classical multiplication kernel associated with an integrable system and
discuss its quantization. Finally, we discuss geometric and algebraic aspects
of method of separation of variables, and describe hypothetically a cyclic
$D$-module for the generalized multiplication kernels for Hitchin systems for
groups $GL_r$.
