Integral representation of solution to the non-stationary Lam\'e equation

We consider methods for constructing explicit solutions of the non-stationary Lam\'e equation, which is a generalization of the classical Lam\'e equation, that has appeared in works on integrable models, conformal field theory, high energy physics and representation theory. We also present a general method for constructing integral representations of solutions to the non-stationary Lam\'e equation by a recursive scheme. Explicit integral representations, for special values of the model parameters, are also presented. Our approach is based on kernel function methods which can be naturally generalized to the non-stationary Heun equation.