Let Δ be an affine building of type Ã2 and let A be its fundamental apartment. We consider the set U_0 of vertices of type 0 of A and prove that the Hecke algebra of all W_0-invariant difference operators with constant coefficients acting on U_0 has three generators. This property leads us to define three Laplace operators on vertices of type 0 of Δ. We prove that there exists a joint eigenspace of these operators having dimension greater than |W_0|. This implies that there exist joint eigenfunctions of the Laplacians that cannot be expressed, via the Poisson transform, in terms of a finitely addittive measure on the maximal boundary Ω of Δ.