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.