A normal form of first order partial differential equations with singular solution (Q1913564)

The author deals with singular solutions of the equation \[ F(x_1,\dots, x_n, z, p_1,\dots, p_n)= 0\quad (p_i= \partial z/\partial x_i), \] that is, with solutions (generalized in the Lie sense) which cannot be included into any complete solution. Denoting by \(\Sigma_c\subset J^1(\mathbb{R}^n, \mathbb{R})\) the singular subset consisting of all points where the contact form is proportional to the differential \(dF\), such a solution exists if and only if \(\Sigma_c\) is an \(n\)-dimensional submanifold. Then there is a contact diffeomorphism germ \(f\) with the property \(f(F^{- 1}(0))= \{y= 0\}\).