Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 115 / 2 / ii / Solution

Using the left-invariant frame of the real Heisenberg group, horizontality givesFor an arbitrary starting point $(x_0,y_0,z_0)$,This is the control equation for the Heisenberg horizontal distribution. The final coordinate records a signed area-like integral, so changing the path in the $(x,y)$ plane can change the endpoint in the central direction.