Density of an individual solution and ergodicity of a family of solutions to the equation $d\eta/d\xi=P(\xi,\,\eta)/Q(\xi,\,\eta)$

The paper provides a sharpened proof of M. G. Khudai–Verenov's theorem on the density in $C^2$ of solutions to the equation $d\eta/d\xi=F/Q$ on condition that this equation has two singular points at infinity whose characteristic numbers satisfy certain constraints of the incommensurability type.