Real moduli in local classification of Goursat flags (Q1775388)

The Goursat flags are sequences of subbundles \(D^r\subset D^{r-1}\subset\cdots\subset D^0= TM\) \((2\leq r\leq n-2)\) in the tangent bundle \(TM\) of a smooth \(n\)-dimensional manifold \(M\) such that \(\text{corank\,}D^l= 1\) and \([D^l, D^l]= D^{l-1}\) \((r\leq l< 0)\). The paper is devoted to thorough analysis of the Kumpera-Ruiz pseudo-normal form: around any point \(p\in M\), \(D^l\) has a basis \(\omega^1,\dots, \omega^l\) such that \(\omega^1= dx^2- x^3 dx^1\), \(\omega^2= dx^3- x^4 dx^1\), \(\omega^3= dx^{i_3}- x^5 dx^{j_3}\) (either \(i_3= 4\), \(j_3= 1\) or \(i_3= 1\), \(j_3= 4\)), \(\omega^4= dx^{i_4}- (c^6+ x^6) dx^{j_4}\) (either \(i_4= 5\), \(j_4= j_3\) or \(i_4= j_3\), \(j_4= 5\)), \(\dots\), \(\omega^r= dx^{i_r}- (c^{r+2}+ x^{r+2}) dx^{j_r}\) (either \(i_r= r+1\), \(j_r= j_{r-1}\) or \(i_r= j_{r-1}\), \(i_r= r+1\)).