  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.
