$\mathcal{P}\mathcal{T}$ deformation of Calogero-Sutherland models

Calogero-Sutherland models of $N$ identical particles on a circle are deformed away from hermiticity but retaining a $\mathcal{P}\mathcal{T}$ symmetry. The interaction potential gets completely regularized, which adds to the energy spectrum an infinite tower of previously non-normalizable states. For integral values of the coupling, extra degeneracy occurs and a nonlinear conserved supersymmetry charge enlarges the ring of Liouville charges. The integrability structure is maintained. We discuss the $A_{N-1}$-type models in general and work out details for the cases of $A_2$ and $G_2$.