For Lie groups admitting a commutative ideal h with h perpendicular contained in h, the Laplace-Beltrami equation reduces via the orbit method to a first-order linear PDE with explicit solutions and nonlocal symmetries.
An integrable class of differential equations with nonlocal nonlinearity on Lie groups,
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Pseudo-Riemannian Lie algebras with coisotropic ideals and integrating the Laplace-Beltrami equation on Lie groups
For Lie groups admitting a commutative ideal h with h perpendicular contained in h, the Laplace-Beltrami equation reduces via the orbit method to a first-order linear PDE with explicit solutions and nonlocal symmetries.