It is a result of Jouanolou's that a complex analytic foliation of proper codimension on an open subset of the projective space has no compact leaves. Thus, for a singular algebraic foliation of the projective space, a closed subvariety which is generically a leaf must contain singularities of the foliation. Using techniques different from Jouanolou's, Steven Kleiman and I show that also algebraic solutions to a Pfaff system of equations, be the system completely integrable or not, must contain singularities of the system.
We apply our results to a problem considered by Darboux and Poincare, that of finding conditions for the existence of rational first integrals of algebraic planar differential systems. We recover and improve related bounds on Tjurina numbers found recently by Du Plessis and Wall.
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