Let \(M\) be a \(C^\infty\)-manifold of dimension \(n\) and let \(E\) be a \(k\)-distribution on \(M\). Let \(E_1=E\), \(E_i=[E,E_{i-1}]=\{[X,Y];\;X\in E\;\text{and}\;Y\in E_{i-1}\}\) and \(E^1=E\), \(E^i=[E^{i-1},E^{i-1}]=\{[X,Y];\;X, Y\in E^{i-1}\}\) for \(i\geq 2\). Let \(r_i(p)=\dim E_i(p)\) and \(m_i(p)=\dim E^i(p)\) for every point \(p\in M\). The sequence \([r_1(p), r_2(p),\dots]\) (resp. \([m_1(p), m_2(p),\dots]\)) is called the small growth vector \((s.g.v)_p\) (resp. big growth vector \((b.g.v)_p\)) of \(E\) at \(p\in \mathbb R^n\). A \(k\)-distribution \(E\) is said to satisfy the Goursat condition at a point \(p\in M\) if \((b.g.v)_p = [k, k + 1, k + 2, \dots, n]\). In this case, the annihilator \(E^\perp\) of \(E\) is called a Goursat system. A path \(\gamma: [\alpha,\beta]\subset \mathbb R\to M\) is said to be \(E\)-admissible if for any \(t\in [\alpha,\beta]\), \(\gamma(t)\in E(\gamma(t))\) and the derivative \(\gamma'(t)\) exists. Let \(\Omega_a([\alpha,\beta])=\{\gamma;\;\gamma: [\alpha,\beta]\to M,\;\text{is }E\)-admissible and