  We discuss a Lie algebraic and differential geometry construction of
solutions to some multidimensional nonlinear integrable systems describing
diagonal metrics on Riemannian manifolds, in particular those of zero and
constant curvature. Here some special solutions to the Lam\'e and Bourlet type
equations, determining by n arbitrary functions of one variable are obtained in
an explicit form. For the case when the sum of the diagonal elements of the
metric is a constant, these solutions are expressed as a product of the Jacobi
elliptic functions and are determined by 2n arbitrary constants.
