  We study the system $-\Delta \mathbf{u}=| \mathbf{u}|^{\alpha-1} \mathbf{u}$
with $1<\alpha\leq\frac{n+2}{n-2}$, where $ \mathbf{u}=(u_1,\dots,u_m)$, $m\geq
1$, is a $C^2$ nonnegative function that develops an isolated singularity in a
domain of $\mathbb{R}^n$, $n\geq 3$. Due to the multiplicity of the components
of $ \mathbf{u}$, we observe a new Pohozaev invariant other than the usual one
in the scalar case, and also a new class of singular solutions provided that
the new invariant is nontrivial. Aligned with the classical theory of the
scalar equation, we classify the solutions on the whole space as well as the
punctured space, and analyze the exact asymptotic behavior of local solutions
around the isolated singularity. On the technical level, we adopt the method of
the moving spheres and the balanced-energy-type monotonicity functionals.
