  The purpose of this brief paper is to prove De Giorgi type results for stable
solutions of the following nonlocal system of integral equations in two
dimensions $$ L(u_i) = H_i(u) \quad \text{in} \ \ \mathbb R^2 , $$ where
$u=(u_i)_{i=1}^m$ for $u_i: \mathbb R^n\to \mathbb R$, $H=(H_i)_{i=1}^m$ is a
general nonlinearity. The operator $L$ is given by $$L(u_i (x)):= \int_{\mathbb
R^2} [u_i(x) - u_i(z)] K(z-x) dz,$$ for some kernel $K$. The idea is to apply a
linear Liouville theorem for the quotient of partial derivatives, just like in
the proof of the classical De Giorgi's conjecture in lower dimensions. Since
there is no Caffarelli-Silvestre local extension problem associated to the
above operator, we deal with this problem directly via certain integral
estimates.
