On generic differential SOn-extensions. (English summary) Proc. Amer. Math. Soc. 136 (2008), no. 4, 1145–1153. Summary: “Let C be an algebraically closed field with trivial derivation and let F denote the differential rational field C〈Yij〉, with Yij, 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n, i ≤ j, differentially independent indeterminates over C. We show that there is a Picard-Vessiot extension E ⊃ F for a matrix equation X ′ = XA(Yij), with differential Galois group SOn, with the property that if F is any differential field with field of constants C, then there is a Picard-Vessiot extension E ⊃ F with differential Galois group H ≤ SOn if and only if there are fij ∈ F with A(fij) well defined and the equation X ′ = XA(fij) giving rise to the extension E ⊃ F.
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