On Equation for Initial Values in Theory of the Second Order Ordinary   Differential Equations
  We consider the properties of the second order nonlinear differential equations b''= g(a,b,b') with the function g(a,b,b'=c) satisfying the following nonlinear partial differential equation $$   \frac{d^2 g_{cc}}{da^2}-g_{c}\frac{dg_{cc}}{da}-4\frac{dg_{bc}}{da}+ $$ $$   +4g_{c}g_{bc}-3g_{b}g_{cc}+6g_{bb}=0, $$ where: $$ \frac{d}{da}=\frac{\partial}{\partial a}+c \frac{\partial}{\partial b}+ g \frac{\partial}{\partial c}. $$   Any equation b''=g(a,b,b') with this condition on function g(a,b,b') has the General Integral F(a,b,x,y)=0 shared with General Integral of the second order ODE's y''=f(x,y,y') with condition $\frac{\partial^4 f}{\partial y'^4}=0$ on function f(x,y,y') or $$ y''+a_{1}(x,y)y'^3+3a_{2}(x,y)y'^2+3a_{3}(x,y)y'+a_{4}(x,y)=0 $$ with some coefficients a_{i}(x,y). 
Valerii S. Dryuma, Maxim Pavlov