In this paper, we classify Möbius invariant differential operators of second order in two-dimensional Euclidean space, and establish a Liouville type theorem for general Möbius invariant elliptic equations. The equations are naturally associated with a continuous family of convex cones Γp$$\Gamma _p$$ in R2$$\mathbb R^2$$, with parameter p∈[1,2]$$p\in [1, 2]$$, joining the half plane Γ1:={(λ1,λ2):λ1+λ2>0}$$\Gamma _1:=\{ (\lambda _1, \lambda _2):\lambda _1+\lambda _2>0\}$$ and the first quadrant Γ2:={(λ1,λ2):λ1,λ2>0}$$\Gamma _2:=\{ (\lambda _1, \lambda _2):\lambda _1, \lambda _2>0\}$$. Chen and C. M. Li established in 1991 a Liouville type theorem corresponding to Γ1$$\Gamma _1$$ under an integrability assumption on the solution. The uniqueness result does not hold without this assumption. The Liouville type theorem we establish in this paper for Γp$$\Gamma _p$$, 1