Uniformity of $cc$-balls on some class of 2-step Carnot groups

For some class of 2-step Carnot groups $\Bbb H_{\alpha_1,\dots,\alpha_n}^1$ that includes Heizenberg groups we proved that Carnot-Carathéodory balls ($cc$-balls) of these groups are uniform domains. We studied the geometry of the set of points of $\Bbb H_{\alpha_1,\dots,\alpha_n}^1$ joined with identity element of $\Bbb H_{\alpha_1,\dots,\alpha_n}^1$ more than one Carnot-Carathéodory $cc$- shortest path.