# The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories

## TL;DR

This paper analyzes the algebraic and geometric properties of the maximally superintegrable Perlick system type I, revealing how parameters influence the system's trajectories and symmetries.

## Contribution

It provides a comprehensive algebraic and geometric analysis of the Perlick system type I, including constants of motion and their Poisson algebra for all parameter values.

## Key findings

- Trajectories depend on the sign of parameter K, on compactness of manifolds.
- The Poisson algebra of constants of motion is fully derived.
- The rational parameter β significantly influences the system's behavior.

## Abstract

In this paper, we investigate the main algebraic properties of the maximally superintegrable system known as "Perlick system type I". All possible values of the relevant parameters, $K$ and $\beta$, are considered. In particular, depending on the sign of the parameter $K$ entering in the metrics, the motion will take place on compact or non compact Riemannian manifolds. To perform our analysis we follow a classical variant of the so called factorization method. Accordingly, we derive the full set of constants of motion and construct their Poisson algebra. As it is expected for maximally superintegrable systems, the algebraic structure will actually shed light also on the geometric features of the trajectories, that will be depicted for different values of the initial data and of the parameters. Especially, the crucial role played by the rational parameter $\beta$ will be seen "in action".