# SO(4)-symmetry of mechanical systems with 3 degrees of freedom

## TL;DR

This paper investigates the existence of mechanical systems with three degrees of freedom that have a specific symmetry algebra, providing a new example and showing that not all such systems possess this symmetry.

## Contribution

The authors present a new mechanical system with 3 degrees of freedom exhibiting o(4) symmetry, expanding understanding beyond the Coulomb system.

## Key findings

- Existence of a non-centrally symmetric system with o(4) symmetry
- Not all 3-degree-of-freedom systems have o(4) Lie algebra symmetry
- Provided explicit construction of such a system

## Abstract

We answered the old question: does there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra o(4) by means of the Poisson brackets? We presented a system which is not centrally symmetric, but has such 6 first integrals. We showed also that not every mechanical system with 3 degrees of freedom possesses such Lie algebra o(4).