  Consider the following coupled elliptic system of equations
  \begin{equation*} \label{}
  (-\Delta)^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} \
\ \mathbb{R}^n ,
  \end{equation*}
  where $0<s\le 2$, $p>1$, $m\ge1$, $u=(u_i)_{i=1}^m$ and $u_i:\mathbb R^n\to
\mathbb R$. The qualitative behavior of solutions of the above system has been
studied from various perspectives in the literature including the free boundary
problems and the classification of solutions. For the case of local scalar
equation, that is when $m=1$ and $s=1$, Gidas and Spruck in \cite{gs} and later
Caffarelli, Gidas and Spruck in \cite{cgs} provided the classification of
solutions for Sobolev sub-critical and critical exponents. More recently, for
the case of local system of equations that is when $m\ge1$ and $s=1$ a similar
classification result is given by Druet, Hebey and V\'{e}tois in \cite{dhv} and
references therein. In this paper, we derive monotonicity formulae for entire
solutions of the above local, when $s=1,2$, and nonlocal, when $0<s<1$ and
$1<s<2$, system. These monotonicity formulae are of great interests due to the
fact that a counterpart of the celebrated monotonicity formula of
Alt-Caffarelli-Friedman \cite{acf} seems to be challenging to derive for system
of equations. Then, we apply these formulae to give a classification of finite
Morse index solutions. In the end, we provide an open problem in regards to
monotonicity formulae for Lane-Emden systems.
