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

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.