  We study a Lax pair in a $2$-parameter Lie algebra in various
representations. The overlap coefficients of the eigenfunctions of $L$ and the
standard basis are given in terms of orthogonal polynomials and orthogonal
functions. Moreover, eigenfunctions for the operator $L$ for a Lax pair for
$\mathfrak{sl}(d+1,\mathbb C)$ are studied in certain representations.
