Transcendental constants over the coefficient fields in differential elliptic function fields

Let k be a differential field of characteristic 0, and Ω be a universal extension of k. Suppose that the field of constants k₀ of k is algebraically closed. Consider the following differential polynomial of the first order over k in a single indeterminate y: T(y) = (y′)² - λS (y; k) λ ∈ k; λ ≠ 0; here S(y; k) = y(1 - y)(1 - κ²y); κ ∈ k; k² ≠ 0, 1; k′ = 0. Take a generic point z of the general solution of T. Then, z is transcendental over k, and k(z, z′) is called a differential elliptic function field. We prove the following: THEOREM. Let k(z, z′) be a differential elliptic function field over k. Then, there exists a finitely generated differential extension field k* of k such that the following three conditions are satisfied: (i) z is transcendental over k*; (ii) the field of constants of k* is the same as k₀; (iii) there exists an element ζ of Ω such that k*(z, z′) = k*(ζ, ζ′) and (ζ′)² = 4S(ζ; κ) with the same modulus as κ.