The construction of first-order PDE from a complete integral begins with a family of surfaces. The main idea is simple. We start with a relation containing two arbitrary parameters. Then we differentiate it with respect to $x$ and $y$. After that, we eliminate the two parameters. The remaining equation contains $x,y,z,p,q$ only. This is the required first-order partial differential equation.

Let a two-parameter family of surfaces be given by $$\begin{equation*} f(x,y,z,a,b)=0, \quad \text{(1)} \end{equation*}$$ where $a$ and $b$ are arbitrary parameters. Also let $$\begin{equation*} p=\frac{\partial z}{\partial x}, \qquad q=\frac{\partial z}{\partial y}. \end{equation*}$$ We differentiate (1) partially with respect to $x$ and $y$. Since $z$ is a function of $x$ and $y$, the chain rule gives $$\begin{align*} f_x+p f_z &=0, \nonumber \\ f_y+q f_z &=0. \quad \text{(2)} \end{align*}$$ Now equations (1) and (2) contain the two arbitrary parameters $a$ and $b$. If possible, we eliminate $a$ and $b$ from these equations. Then we get an equation of the form $$\begin{equation*} F(x,y,z,p,q)=0. \quad \text{(3)} \end{equation*}$$ Equation (3) is a first-order partial differential equation.