Various aspects of differential equations having a complete set of independent first integrals

TL;DR

This paper investigates differential equations in multi-dimensional spaces that possess a complete set of independent first integrals, focusing on specific quadratic forms involving constants and variables.

Contribution

It characterizes differential equations with a complete set of independent first integrals, especially those with integrals of a particular quadratic form involving constants and variables.

Findings

Identification of conditions for differential equations to have a complete set of first integrals.

Explicit form of first integrals involving quadratic expressions and constants.

Analysis of the structure of such differential equations in multi-dimensional spaces.

Abstract

In this paper we study the differential equations in $D\subseteq \R^{2N}$ having a complete set of independent first integrals. In particular we study the case when the first integrals are \[f_\nu=(Ax_\nu+By_\nu)^2+\displaystyle\sum_{j=1}^{N}\dfrac{(x_\nu y_j-x_jy_\nu)^2}{a_\nu-a_j},\]for $\nu=1,...,N,$ where $A,B$ and $a_1<a_2...<a_N$ are constants.