  The O(N) invariant quartic anharmonic oscillator is shown to be exactly
solvable if the interaction parameter satisfies special conditions. The problem
is directly related to that of a quantum double well anharmonic oscillator in
an external field. A finite dimensional matrix equation for the problem is
constructed explicitly, along with analytical expressions for some excited
states in the system. The corresponding Niven equations for determining the
polynomial solutions for the problem are given.
