# Intertwining operator for $AG_2$ Calogero-Moser-Sutherland system

## TL;DR

This paper constructs an explicit intertwining operator linking a generalized Calogero-Moser-Sutherland Hamiltonian associated with the combined $A_2$ and $G_2$ root systems to the standard $G_2$ Hamiltonian, proving its integrability.

## Contribution

It introduces a new intertwining operator for a combined root system Hamiltonian, demonstrating its integrability through explicit quantum integral construction.

## Key findings

- Established integrability of the generalized Hamiltonian.
- Constructed explicit intertwining operator.
- Derived quantum integral of order 6.

## Abstract

We consider generalised Calogero-Moser-Sutherland quantum Hamiltonian $H$ associated with a configuration of vectors $AG_2$ on the plane which is a union of $A_2$ and $G_2$ root systems. The Hamiltonian $H$ depends on one parameter. We find an intertwining operator between $H$ and the Calogero-Moser-Sutherland Hamiltonian for the root system $G_2$. This gives a quantum integral for $H$ of order 6 in an explicit form thus establishing integrability of $H$.