General solution of the Zero Gaussian Curvature Monge-Ampère Differential Equation

Question

Suppose we have (but we don't know what it is) a parameterized surface of Monge type $$ \textbf{x}(u,v)=\bigl(u,v,f(u,v)\bigr).$$ The particular Monge-Ampère equation is $$ \boxed{f_{uu} f_{vv}-(f_{uv})^2=0}.$$ I know that the surfaces with $K=0$ verify the equation, like planes, cylinders, ... According to the theory of Differential Equations (Moon and Spencer, 1969; Zwillinger, 1997) this particular Monge-Ampère equation verify: $$ \text{d}f= f_u \text{d}u + f_v \text{d}v, \quad \text{d}f_u =0, \quad \text{d}f_v=0.$$ What this result does mean? Can I obtain an explicit expression of $\textbf{x}(u,v)$?