Integrability of and differential–algebraic structures for spatially 1D hydrodynamical systems of Riemann type

•A new differential–algebraic–geometric approach for testing integrability is described.•The approach is applied to a generalized Riemann type hydrodynamic system.•The approach is applied to a generalized Ostrovsky–Vakhnenko system.•The approach is applied to a new two-component Burgers type hydrodynamic system.

A differential–algebraic approach to studying the Lax integrability of a generalized Riemann type hydrodynamic hierarchy is revisited and a new Lax representation is constructed. The related bi-Hamiltonian integrability and compatible Poissonian structures of this hierarchy are also investigated using gradient-holonomic and geometric methods.The complete integrability of a new generalized Riemann hydrodynamic system is studied via a novel combination of symplectic and differential–algebraic tools. A compatible pair of polynomial Poissonian structures, a Lax representation and a related infinite hierarchy of conservation laws are obtained.In addition, the differential–algebraic approach is used to prove the complete Lax integrability of the generalized Ostrovsky–Vakhnenko and a new Burgers type system, and special cases are studied using symplectic and gradient-holonomic tools. Compatible pairs of polynomial Poissonian structures, matrix Lax representations and infinite hierarchies of conservation laws are derived.