# $\mathbb{CP}^N$-Rosochatius system, superintegrability, supersymmetry

## TL;DR

This paper introduces a new superintegrable $ ext{CP}^N$-Rosochatius system with magnetic coupling, analyzing its symmetries, solutions, and supersymmetric extensions, expanding understanding of integrable models on complex projective spaces.

## Contribution

It presents a novel superintegrable system on $ ext{CP}^N$ with magnetic fields, detailing its constants of motion, algebra, and supersymmetric extensions, linking it to known oscillator models.

## Key findings

- Derived constants of motion and their algebra.
- Provided classical and quantum solutions.
- Constructed supersymmetric extensions under specific conditions.

## Abstract

We propose new superintegrable mechanical system on the complex projective space $\mathbb{CP}^N$ involving a potential term together with coupling to a constant magnetic fields. This system can be viewed as a $\mathbb{CP}^N$-analog of both the flat singular oscillator and its spherical analog known as "Rosochatius system". We find its constants of motion and calculate their (highly nonlinear) algebra. We also present its classical and quantum solutions. The system belongs to the class of "K\"ahler oscillators" admitting $SU(2|1)$ supersymmetric extension. We show that, in the absence of magnetic field and with the special choice of the characteristic parameters, one can construct $\mathcal{N}=4, d=1$ Poinacar\'e supersymmetric extension of the system considered.