  We study a system of nonlinear elliptic PDEs associated with a
quasi-homogeneous polynomial. These equations were proposed by Witten as the
replacement for the Cauchy-Riemann equation in the singularity
(Landau-Ginzburg) setting. We introduce a perturbation to the equation and
construct a virtual cycle for the moduli space of its solutions. Then, we study
the wall-crossing of the deformation of the virtual cycle under perturbation
and match it to classical Picard-Lefschetz theory. An extended virtual cycle is
obtained for the original equation. Finally, we prove that the extended virtual
cycle satisfies a set of axioms similar to those of Gromov-Witten theory and
r-spin theory.
