Metric lines in Engel-type groups and the nilpotent group N6,3,1

Given a sub-Riemannian manifold, which geodesics are "metric lines" (i.e. globally minimizing geodesics)? This article takes the first steps in answering this question for "arbitrary rank" and "non-integrable" Carnot groups. We classify the metric lines of the Engel-type groups $Eng(n)$ (Theorem B) and give a partial classification for the group of four-by-four nilpotent triangular matrices N6,3,1 (Theorem C). The sub-Riamannian structure of the former group is defined on a non-integrable distribution of rank $n+1$ and the geodesic flow of the latter group is not algebraically integrable.