  In this paper we investigate relations between solutions to the minimal
surface equation in Euclidean $3$-space $\mathbb{E}^3$, the zero mean curvature
equation in Lorentz-Minkowski $3$-space $\mathbb{L}^3$ and the Born-Infeld
equation under Wick rotations. We prove that the existence conditions of real
solutions and imaginary solutions after Wick rotations are written by
symmetries of solutions, and reveal how real and imaginary solutions are
transformed under Wick rotations. We also give a transformation theory for zero
mean curvature surfaces containing lightlike lines with some symmetries. As an
application, we give new correspondences among some solutions to the above
equations by using the non-commutativity between Wick rotations and isometries
in the ambient space.
