Notes on generalized special Lagrangian equation

We obtain a prior $C^{1,1}$ estimates for some Hessian (quotient) equations with positive Lipschitz right hand sides, through studying a twisted special Lagrangian equation. The results imply the interior $C^{2,\alpha }$ regularity for $C^0$ viscosity solutions to ${\sigma }_2=f^2(x)$ in dimension 3, with positive Lipschitz $f(x)$.