Vector-valued covariant differential operators for the M\"obius transformation

We obtain a family of functional identities satisfied by vector-valued functions of two variables and their geometric inversions. For this we introduce particular differential operators of arbitrary order attached to Gegenbauer polynomials. These differential operators are symmetry breaking for the pair of Lie groups $(SL(2,\mathbb{C}),SL(2,\mathbb{R}))$ that arise from conformal geometry.