Flag systems and ordinary differential equations (Q1570439)

Denoting by \(DS\) the derived subsystem of a Pfaffian system \(S\) (i.e., \(\omega\in DS\) if and only if \(d\omega=0\bmod S)\), we have a nonascending sequence \(S\supset DS\supset D^2S\supset\cdots\supset D^lS=D^{l+1}S\) of Pfaffian systems terminating with stationarity. Then \(S\) is called a flag system if \(\text{rank} D^kS=\text{const}-k\) \((k\leq l)\). The author discusses flag systems of the length \(l=2\) or \(l=3\) in the connection to the Monge problem: to resolve an underdetermined system of ordinary equations \((n\) equations, \(m\) unknown functions, \(n<m)\) by means of explicit formulae (involving \(m-n\) arbitrary functions). The problem was resolved by E. Cartan (for the case \(m=n+1)\) and the article provides a beautiful introduction into this unfortunately forgotten area. (Reviewer's remark: the really difficult cases when \(m>n+1\) are always tacitly passed over without any comments).