Extension of the shooting argument to nonzero subcritical phases

Determine whether a shooting argument performed directly on the full interval [-1,1], without using odd reflection, can succeed for constant-phase special Lagrangian equations with nonzero subcritical phases.

Background

The paper constructs a singular graphical special Lagrangian cone by reducing the zero-phase equation to an ordinary differential equation and extending a solution across the parameter value s=0 by odd reflection. This reflection relies on the zero-phase symmetry: replacing the Hessian by its negative preserves the phase-zero equation but generally changes a nonzero constant phase.

For nonzero subcritical phases, the authors do not establish whether the reduced shooting problem can instead be solved on the entire interval [-1,1] without invoking reflection. Such an extension could potentially produce analogous singular graphical special Lagrangian solutions for other phases, but the paper leaves this question unresolved.