A Calogero Model with root string representatives of infinite order Coxeter orbits

We present a worked example for the new extensions of the multi-particle Calogero model endowed with infinite Weyl group symmetry of affine and hyperbolic type. Building upon the hyperbolic extension of the $A_3$-Kac-Moody algebra, we construct an explicit realisation of the model in terms of infinite root systems generated from Coxeter orbits. To address the challenge of summing over infinitely many roots, we introduce root string representatives that span the invariant root space while preserving invariance under the affine Weyl group. This approach yields closed-form expressions for the potentials, which by construction are invariant under the full affine Weyl symmetry. Moreover, we demonstrate that in an appropriate infinite-coordinate limit the model reduces smoothly to the conventional four particle $A_3$-Calogero system. Our construction constitutes a systematic method for implementing infinite-dimensional symmetries into Calogero-type models, thus broadening their algebraic and physical applicability.