Cubeintegrability

Cubeintegrability is a concept in mathematical physics, specifically within the study of completely integrable systems. A classical or quantum mechanical system is considered cubeintegrable if its dynamics can be solved using a method that involves a certain type of "cube" or a related mathematical structure. This often refers to the existence of a sufficient number of conserved quantities that are in involution, meaning they commute with each other under the Poisson bracket. These conserved quantities allow for the explicit integration of the equations of motion.

The term "cubeintegrable" is not as universally standardized as "integrable" itself, but it generally implies a higher degree of integrability, often related to specific algebraic structures or the presence of Lax pairs with particular properties. In some contexts, it might refer to systems that can be linearized on an Abelian variety or systems exhibiting specific algebraic properties that simplify their analysis. The precise mathematical definition can vary depending on the author and the specific field of research. The goal of identifying cubeintegrable systems is to find systems whose complex dynamics can be completely understood and solved analytically.