Symmetry and separation of variables for the Hamilton-Jacobi equation W_t²-W_x²-W_y² = 0

We present a detailed group theoretical study of the problem of separation of variables for the characteristic equation of the wave equation in one time and two space dimensions. Using the well-known Lie algebra isomorphism between canonical vector fields under the Lie bracket operation and functions (modulo constants) under Poisson brackets, we associate, with each R -separable coordinate system of the equation, an orbit of commuting constants of the motion which are quadratic members of the universal enveloping algebra of the symmetry algebra o(3,2). In this, the first of two papers, we essentially restrict ourselves to those orbits where one of the constants of the motion can be split off, giving rise to a reduced equation with a nontrivial symmetry algebra. Our analysis includes several of the better known two-body problems, including the harmonic oscillator and Kepler problems, as special cases.