a complete integral
An ordinary differential equation of order one has a general solution carrying one arbitrary constant; pick the constant and you pick a particular curve. A first-order PDE in two variables is richer, and one of the oldest ways to capture its solutions is the complete integral: a single formula for u that contains two arbitrary constants, enough parameters to bend the solution surface in both independent directions.
Concretely, a complete integral of F(x, y, u, u_x, u_y) = 0 is a family u = phi(x, y, a, b) depending on two parameters a and b, each member of which solves the equation, and which is genuinely two-parameter (the surfaces it gives are not all the same shape slid around). The surprise is that this two-parameter family is not merely a pile of special solutions — it is a generating set. By choosing b as a function of a, b = g(a), and then forming the envelope of the resulting one-parameter family (eliminate a between u = phi(x,y,a,g(a)) and the derivative condition phi_a + phi_b g'(a) = 0), you sweep out new solutions, including the so-called general integral built around arbitrary data. The envelope of the full two-parameter family gives yet another, the singular integral.
This is the classical Charpit-Jacobi theory, and although the modern method of characteristics largely supersedes it, the complete integral remains the cleanest way to see how much freedom a first-order PDE has: two constants, then envelopes promote them to an arbitrary function — the hallmark that a PDE's solution set is parametrized by functions, not just numbers. It is also the route by which Hamilton-Jacobi theory turns a complete integral into the full solution of a mechanics problem.
The eikonal equation u_x^2 + u_y^2 = 1 has the complete integral u = a x + sqrt(1 - a^2) y + b, a two-parameter family of tilted planes (each a flat wavefront). Taking envelopes of sub-families of these planes builds curved solutions such as the cone u = sqrt(x^2 + y^2), the distance from a point.
Two constants give a family of flat solutions; their envelopes promote those constants into an arbitrary function.
A complete integral is not the most general solution: it has only two constants, whereas the full solution set of a first-order PDE involves an arbitrary function. The link is the envelope operation, which trades a discrete parameter for a functional one — without it, two constants would badly undercount the solutions.