  This is the second of three papers math.DG/0111324, math.DG/0204343 studying
special Lagrangian 3-submanifolds (SL 3-folds) N in C^3 invariant under the
U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g,
using analytic methods. The three papers are surveyed in math.DG/0206016.
  If N is such a 3-fold then |z_1|^2-|z_2|^2=2a on N for some real number a.
Locally, N can be written as a kind of graph of functions u,v : R^2 --> R
satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is
like a holomorphic function of x+iy.
  The first paper math.DG/0111324 studied the case when a is nonzero. Then u,v
are smooth and N is nonsingular. It proved existence and uniqueness for
solutions of two Dirichlet problems derived from the equations on u,v. This
implied existence and uniqueness for a large class of nonsingular
U(1)-invariant SL 3-folds in C^3, with boundary conditions.
  In this paper and its sequel math.DG/0204343 we focus on the case a=0. Then
the nonlinear Cauchy-Riemann equation is not always elliptic. Because of this
there may be points (x,0) where u,v are not differentiable, corresponding to
singular points of N. This paper is concerned largely with technical analytic
issues, and the sequel with the geometry of the singularities of N.
  We prove a priori estimates for derivatives of solutions of the nonlinear
Cauchy-Riemann equation, and use them to show existence and uniqueness of weak
solutions u,v to the two Dirichlet problems when a=0, which are continuous and
weakly differentiable. This gives existence and uniqueness for a large class of
singular U(1)-invariant SL 3-folds in C^3, with boundary conditions.
