  A detailed study of solutions to the first order partial differential
equation H(x,y,z_x,z_y)=0, with special emphasis on the eikonal equation
z_x^2+z_y^2=h(x,y), is made near points where the equation becomes singular in
the sense that dH=0, in which case the method of characteristics does not
apply. The main results are that there is a strong lack of uniqueness of
solutions near such points and that solutions can be less regular than both the
function H and the initial data of the problem, but that this loss of
regularity only occurs along a pair of curves through the singular point. The
main tools are symplectic geometry and the Sternberg normal form for
Hamiltonian vector fields.
