Classification of minimal Lorentzian surfaces in S⁴₂(1) with constant Gaussian and normal curvatures

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In this paper we consider Lorentzian surfaces in the 4-dimensional pseudo-Riemannian sphere S⁴₂(1) with index 2 of curvature one. We obtain the complete classification of minimal Lorentzian surfaces S⁴₂(1) whose Gaussian and normal curvatures are constants. We conclude that such surfaces have the Gaussian curvature 1/3 and the absolute value of normal curvature 2/3. We also give some explicit examples.