Symmetries and conservation laws for the generalized $n$-dimensional Ermakov system

TL;DR

This paper revises and completes the classification of symmetries and conservation laws for the generalized n-dimensional Hamiltonian Ermakov system, correcting previous inaccuracies and extending results to higher dimensions.

Contribution

It provides the correct classification of symmetries for the three-dimensional system and extends these findings to the generalized n-dimensional case.

Findings

Corrected previous classification errors

Extended symmetry classification to n-dimensional systems

Established links between background space and dynamics

Abstract

We revise recent results on the classification of the generalized three-dimensional Hamiltonian Ermakov system. We show that a statement published recently is incorrect, while the solution for the classification problem was incomplete. We present the correct classification for the three-dimensional system by using results which related the background space with the dynamics. Finally, we extend our results for the generalized $n$% -dimensional Hamiltonian Ermakov system.