Hierarchies of Manakov–Santini Type by Means of Rota–Baxter and Other Identities

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.