I will consider a system of two coupled Liouville equations on the plane. The system admits so-called scalar solutions, namely such that the two components coincide. These solutions actually solve a scalar Liouville equation on the plane, hence they are very well known and they have been completely classified. On the other hand, much less is known about non-scalar solutions. Using bifurcation theory, I will show the existence of some branches of (non-scalar) solutions bifurcating from a scalar solution.