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

In this paper we study the differential equations in $D\subseteq {\mathbb{R}}^{2N}$ having a complete set of independent first integrals. In particular we study the case when the first integrals are $$f_{\nu }=(A{x}_{\nu }+B{y}_{\nu }{)}^2+{\displaystyle \sum _{j=1}^N{\displaystyle \frac{(x_{\nu }{y}_j-x_j{y}_{\nu }{)}^2}{a_{\nu }-a_j}},}$$for $\nu =1,\mathrm{.}\mathrm{.}\mathrm{.},N,$ where $A,B$ and $a_1<a_2\mathrm{.}\mathrm{.}\mathrm{.}<a_N$ are constants.