Group Analysis of Differential Equations and Integrable Systems − 2018 Kouichi Takemura (Chuo University, Tokyo, Japan) On $q$-deformations of the Heun equation Abstract: The $q$-Heun equation and its variants arise as degenerations of Ruijsenaars-van Diejen operators with one particle. We investigate local properties of these equations. In particular we characterize the variants of the $q$-Heun equation by using analysis of regular singularities. We also consider the quasi-exact solvability of the $q$-Heun equation and its variants. Namely we investigate finite-dimensional subspaces which are invariant under the action of the $q$-Heun operator or variants of the $q$-Heun operator. This is based on arXiv:1712.09564[math.CA].