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In this paper we investigate relations between solutions to the minimalsurface equation in Euclidean $3$-space $\mathbb{E}^3$, the zero mean curvatureequation in Lorentz-Minkowski $3$-space $\mathbb{L}^3$ and the Born-Infeldequation under Wick rotations. We prove that the existence conditions of realsolutions and imaginary solutions after Wick rotations are written bysymmetries of solutions, and reveal how real and imaginary solutions aretransformed under Wick rotations. We also give a transformation theory for zeromean curvature surfaces containing lightlike lines with some symmetries. As anapplication, we give new correspondences among some solutions to the aboveequations by using the non-commutativity between Wick rotations and isometriesin the ambient space.

Wick rotations of solutions to the minimal surface equation, the zero mean curvature equation and the Born-Infeld equation