We consider the problem of controlling the nonlinear 5-dimensional systemsthat are induced by horizontal vector fields $ X $ and $ Y $ which together with their commutators generate some Cartan algebradepending linearly on two piecewise constant controls.We also study the properties of solutions to the systems on interpretinga solution asa horizontal $ k $ -broken line $ L_{k} $ on the canonical Cartan group $ 𝕂 $ ,where the segmentsof $ L_{k} $ are segments of integral curves of the vector fields of the form $ aX+bY $ with $ a,b=\operatorname{const} $ .As regards $ 𝕂 $ ,we prove that 4 isthe minimal number $ N_{𝕂} $ such thatevery two points $ u,v\in 𝕂 $ can be joined by some $ L_{k} $ with $ k\leq N_{𝕂} $ .Thus,we obtain the best version of the Rashevskii–Chow theorem on the Cartan group.We also show thatthe minimal number of segments of a closed horizontal broken line on $ 𝕂 $ equals 6.
