Is harmonic oscillator a free particle in disguise?

Let me start with an example. If one considers a free particle motion on two-dimensional plane and projects it onto the radial coordinate, one gets the following Hamiltonian
$$H = frac{p^2}{2}+g^2 frac{1}{ q^2}$$
which is integrable (and known as a one-particle rational Calogero model). Considering free motions on sphere and other (homogeneous) spaces, one is able to discover different integrable systems.

Hence follows my question: does harmonic oscillator (and its generalizations to multiparticle case) represent a free motion on some space?

In my naïveté I thought about a space with metric of the form $ds^2 = dr^2 + dfrac{dtheta^2}{r^2}$, but this is not, of course, a space of constant curvature, and moreover, its Ricci scalar diverges as $R = -frac{4}{r^2}$ at $r=0$. Of course, in one-particle case it doesn’t matter since all systems of searched form are such, but it might matter later.