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We consider a horizontal minimal surface in 5-sphere s0 :M −→ S ⊂ C. The Gauss, Codazzi and Ricci equations for s0 becomes a unified equation called “Tzitéica equation”of elliptic type as follows. ∂z∂zu = e −2u − e, (1) where z = x + √−1y a local coordinate system for a Riemann surface M and u = u(z, z) is a real valued function on M . The induced metric g on M is given by g = 2edzdz. Let <,> be a Hermitian fibre metric on M ×C3 compatible with g. Since s0 is a minimal immersion, we have < s0, s0 >= 1, ∂z∂zs0 = −es0. (2) The horizontality of s0 with respect to the Hopf fibration S 5 −→ CP 2 means that < ∂zs0, s0 >= 0, < ∂zs0, s0 >= 0 (3) We also have < s0, ∂zs0 >= 0, < s0, ∂zs0 >= 0. The conformality of s0 means that < ∂zs0, ∂zs0 >= 0, < ∂zs0, ∂zs0 >= 0 (4)

Finite gap solutions for horizontal minimal surfaces of finite type in 5-sphere