Pure Picard-Vessiot extensions with generic properties

Given a connected linear algebraic group G over an algebraically closed field C of characteristic 0, we construct a pure Picard-Vessiot extension for G, namely, a Picard-Vessiot extension E superset of F, with differential Galois group G, such that E and F are purely differentially transcendental over C. The differential field E is the quotient field of a G-stable proper differential subring R with the property that if F is any differential field with field of constants C and E superset of F is a Picard-Vessiot extension with differential Galois group a connected subgroup H of G, then there is a differential homomorphism phi : R --> E such that E is generated over F as a differential field by phi( R).