# Proving Compatibility of two Partial differential equation

Given two PDE(s): $F(x,y,z,p,q)=0$
and $G(x,y,z,p,q)=0$
In I.A.N Sneddon's "Elements of Partial Differential Equations",If every solution of $F=0$ is a solution of $G=0$,then $F=0$ and $G=0$ are said to be compatible. But according to another textbook if $F=0$ and $G=0$ have atleast one common solution then they are said to be compatible.So here my question is:
What is the actual "definition of compatibility of two pde(s)"?

Now if I assume that $\left|\frac{\partial(F,G)}{\partial(p,q)}\right|\ne0$ and solve the system $F=0$ and $G=0$ for $p$ and $q$,suppose that I get $p=\phi(x,y,z)$ and $q=\psi(x,y,z)$ so writing complete differential for $dz$,
$dz=pdx +qdy$ ,so this Pfaffian equation is integrable?(This condition has been used in textbook to deduce the condition for two partial differential equations to be compatible).
My second question is:
How does compatibility of "$F=0$ and $G=0$" plus the condition $\left|\frac{\partial(F,G)}{\partial(p,q)}\right|\ne0$ imply integrability of Pfaffian equation "$dz=pdx +qdy$" ?
Please help me know answers to my questions.
Your thoughts are always appreciated,as always.
PS:$p=z_x,q=z_y$