In their 1996 paper, Beals, Gaveau and Greiner found the fundamental solution to a
2
2
-Laplace-type equation in a class of sub-Riemannian spaces. This solution is related to the well-known fundamental solution to the
p
\texttt {p}
-Laplace equation in Grushin-type spaces and the Heisenberg group. We extend the
2
2
-Laplace-type equation to a
p
\texttt {p}
-Laplace-type equation. We show that the obvious generalization does not have desired properties, but rather, our generalization preserves some natural properties.