  The family F of all potentials V(x) for which the Hamiltonian H in one space
dimension possesses a high order Lie symmetry is determined. A sub-family F',
which contains a class of potentials allowing a realization of so(2,1) as
spectrum generating algebra of H through differential operators of finite
order, is identified. Furthermore and surprisingly, the families F and F' are
shown to be related to the stationary KdV hierarchy. Hence, the "harmless"
Hamiltonian H connects different mathematical objects, high order Lie symmetry,
realization of so(2,1)-spectrum generating algebra and families of nonlinear
differential equations. We describe in a physical context the interplay between
these objects.
