= Smooth horizontal reachability in the real Heisenberg group
Every point $(P_x,P_y,P_z)$ can be reached from the identity by a smooth curve tangent to the <Heisenberg horizontal distribution>. Put $a=2(P_z-P_xP_y/2+P_x)$ and choose
$$
\alpha(t)=P_x+a\cos(2\pi t),\qquad
\beta(t)=P_y+2\pi\sin(2\pi t).
$$
Their integrals give $x(1)=P_x$, $y(1)=P_y$, and $z(1)=\int_0^1x(t)\beta(t)dt=P_xP_y/2-P_x+a/2=P_z$. The third coordinate measures the effect of the noncommuting horizontal directions; integrability would instead confine horizontal curves to a two-dimensional leaf.
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