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The exchange operator formalism in polar coordinates, previously consideredfor the Calogero-Marchioro-Wolfes problem, is generalized to a recentlyintroduced, 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 somedifferential-difference operators $D_r$ and $D_{\varphi}$. The latter serve toconstruct $D_{2k}$-extended and invariant Hamiltonians $\chh_k$, from which thestarting Hamiltonians $H_k$ can be retrieved by projection in the $D_{2k}$identity representation space.

Exchange operator formalism for an infinite family of solvable and integrable quantum systems on a plane