During last decades integrable systems helped to answer a number of difficult outstanding questions in mathematics. They led to advances in knot theory, string theory, theory of differential equations, etc. This project consider connections of integrable systems to combinatorial problems and special functions. It is natural to consider quantum spins systems with discrete fluctuating variables as an efficient tool to count ``something" in its configuration space of the system. For example, we can talk about the total number of configurations with 11 spins up, etc. Recently it was found that one can count some special types of matrices using such techniques. It led to a large boost of interest which helped to prove many conjectures in the theory of alternating sign matrices.This project aims to look at different generalizations of such connections.