  The exchange operator formalism in polar coordinates, previously considered
for the Calogero-Marchioro-Wolfes problem, is generalized to a recently
introduced, infinite family of exactly solvable and integrable Hamiltonians
$H_k$, $k=1$, 2, 3,..., on a plane. The elements of the dihedral group $D_{2k}$
are realized as operators on this plane and used to define some
differential-difference operators $D_r$ and $D_{\varphi}$. The latter serve to
construct $D_{2k}$-extended and invariant Hamiltonians $\chh_k$, from which the
starting Hamiltonians $H_k$ can be retrieved by projection in the $D_{2k}$
identity representation space.
