Picard–Vessiot Extensions of Real Differential Fields

For a linear differential equation defined over a formally real differential field $K$ with real closed field of constants $k$, Crespo, Hajto and van der Put proved that there exists a unique formally real Picard–Vessiot extension up to $K$-differential automorphism. However such an equation may have Picard–Vessiot extensions which are not formally real fields. The differential Galois group of a Picard–Vessiot extension for this equation has the structure of a linear algebraic group defined over $k$ and is a $k$-form of the differential Galois group $H$ of the equation over the differential field $K\big(\sqrt{-1}\big)$. These facts lead us to consider two issues: determining the number of $K$-differential isomorphism classes of Picard–Vessiot extensions and describing the variation of the differential Galois group in the set of $k$-forms of $H$. We address these two issues in the cases when $H$ is a special linear, a special orthogonal, or a symplectic linear algebraic group and conclude that there is no general behaviour.