The CMS models have a famous `relativistic´ generalization constructed by Ruijsenaars and van Diejen given by continuous difference equations (also commonly referred to as q-difference equations). These models are among the most complicated in the CMS family and little is known about their exact solutions which would, in many respects, belong to the most general class in the theory of special functions.

A mathematically natural generalization of the CMS type models was discovered by Chalykh, Feigin, Sergeev, and Veselov, and it incorporates two different types of interacting particles. These have a possible interpretation of `particles´ and `holes´ in physics parlance. One goal of this project is to further develop the mathematics related to the deformed RvD type models, such as new special functions, Hilbert space aspects, higher order operators, etc., which is still a developing field of research.

Lax matrices are a fundamental concept in the theory of integrable systems. For the deformed models, the Lax pairs were only known in the trigonometric case only. However, a recent groundbreaking conceptual way of constructing Lax pairs for the CMS and RvD models were found using Dunkl operators. Using Dunkl operators for the deformed models is more complicated and one of the goals is to extend this approach to the deformed models and other generaliztions.

The Painlevé equations are a set of six, special, nonlinear second order differential equations. Remarkably, it is known from the works of many prominent mathematicians that these nonlinear differential equations can be mapped to linear Hamiltonian systems. The Painlevé VI equation, which is the most general, is given by the non-stationary Heun equation (which is part of the one-variable, elliptic CMS family). This is commonly referred to as the Painlevé-Calogero correspondance. This relation is also extend to the multivariate cases were multicomponent generalizations of the Painlevé equations are obtained.