Integrability and conservation laws for two systems of hydrodynamic type derived from the Toda and Volterra systems

We prove that two particular systems of hydrodynamic type can be represented as systems of conservation laws, and that they decouple into non-interacting integrable subsystems. The systems of hydrodynamic type in question were previously constructed, via a matrix partial differential equation, from the Lax pairs for the classical Toda and Volterra systems. The decoupling is guaranteed by the vanishing of the Nijenhuis tensor for each system; integrability of the non-interacting subsystems, thus each system as a whole, is proven for low eigenvalue multiplicities.