  The Lax-Sato approach to the hierarchies of Manakov-Santini type is
formalized in order to extend it to a more general class of integrable systems.
For this purpose some linear operators are introduced, which must satisfy some
integrability conditions, one of them is the Rota-Baxter identity. The theory
is illustrated by means of the algebra of Laurent series, the related
hierarchies are classified and examples, also new, of Manakov-Santini type
systems are constructed, including those that are related to the dispersionless
modified Kadomtsev-Petviashvili equation and so called dispersionless r-th
systems.
