The equation X∇detX=detX∇trX, multiplication of cofactor pair systems, and the Levi-Civita equivalence problem

Cofactor pair systems generalize the separable potential Hamiltonian systems. They admit nn quadratic integrals of motion, they have a bi-Hamilton formulation, they are completely integrable and they are equivalent to separable Lagrangian systems. Cofactor pair systems can be constructed through a peculiar multiplicative structure of the so-called quasi-Cauchy–Riemann equations (cofJ)−1∇V=(cofJ̃)−1∇Ṽ, where JJ and J̃ are special conformal Killing tensors.In this work we have isolated the properties that are responsible for the multiplication, allowing us to give an elegant characterization of systems that admit multiplication. In this characterization the equation X∇detX=detX∇trX plays a central role.We describe how multiplication of quasi-Cauchy–Riemann equations can be considered as a special case of a more general kind of multiplication, defined on the solution space of certain systems of partial differential equations. We investigate algebraic properties of this multiplication and provide several methods for constructing new systems with multiplicative structure. We also discuss the role of the multiplication in the theory of equivalent dynamical systems on Riemannian manifolds, developed by Levi-Civita.