Let $${\mathcal {L}}=\sum _{j=1}^{m}X_{j}^{2}$$ be a Hörmander sum of squares vector fields in $${\mathbb {R}}^{n}$$ , where any $$X_{j}$$ is homogeneous degree 1 with respect to family non-isotropic dilations . Then, {L}}$$ known admit global fundamental solution $$\Gamma (x;y)$$ that can represented as the integral sublaplacian operator on lifting space {R}}^{n}\times {\mathbb {R}}^{p}$$ equi...