Absence of Nontrivial Symmetries to the Heat Equation in Goursat Groups of Dimension at Least 4.

Using the extension method, we study the one-parameter symmetry groups of the heat equation ∂tp = Δp, where Δ = X 1 2 + X 2 2 is the sub-Laplacian constructed by a Goursat distribution span({X1, X2}) in ℝn, where the vector fields X1 and X2 satisfy the commutation relations [X1, Xj] = Xj+1 (where Xn+1 = 0) and [Xj, Xk] = 0 for j ≥ 1 and k ≥ 1. We show that there are no such groups for n ≥ 4 (with exception of the linear transformations of solutions which are admitted by every linear equation). [ABSTRACT FROM AUTHOR]