  Several completely integrable, indeed solvable, Hamiltonian many-body
problems are exhibited, characterized by Newtonian equations of motion
("acceleration equal force"), with linear and cubic forces, in N-dimensional
space (N being an arbitrary positive integer, with special attention to N=2,
namely motions in a plane, and N=3, namely motions in ordinary
three-dimensional space). All the equations of motion are written in covariant
form ("N-vector equal N-vector"), entailing their rotational invariance. The
corresponding Hamiltonians are of normal type, with the kinetic energy
quadratic in the canonical momenta, and the potential energy quadratic and
quartic in the canonical coordinates.
