We prove that almost all points of the Engel group 𝔼α,β are joined with its neutral element by basis horizontal k-broken lines \({L}_{k}^{\varepsilon },k\le 4,\) the links of which are segments of integral lines of basis horizontal left-invariant vector fields X, Y of the Lie algebra of the group 𝔼α,β. Furthermore, there exist elements of the group 𝔼α,β that cannot be joined by basis horizontal k-broken lines \({L}_{k}^{\varepsilon },k\le 4\) .We prove that all points of the Engel group 𝔼α,β are joined with its neutral element by basis horizontal k-broken lines \({L}_{k}^{\varepsilon },k\le 6\) .
