On Generic differential $\operatorname{SO}_n$-extensions

Let $\mathcal C$ be an algebraically closed field with trivial derivation and let $\mathcal F$ denote the differential rational field $\mathcal C\langle Y_{ij}\rangle$, with $Y_{ij}$, $1\leq i\leq n-1$, $1\leq j\leq n$, $i\leq j$, differentially independent indeterminates over $\mathcal C$. We show that there is a Picard-Vessiot extension $\mathcal E\supset \mathcal F$ for a matrix equation $X'=X\mathcal A(Y_{ij})$, with differential Galois group $\operatorname{SO}_n$, with the property that if $F$ is any differential field with field of constants $\mathcal C$, then there is a Picard-Vessiot extension $E\supset F$ with differential Galois group $H\leq\operatorname{SO}_n$ if and only if there are $f_{ij}\in F$ with $\mathcal A(f_{ij})$ well defined and the equation $X'=X\mathcal A(f_{ij})$ giving rise to the extension $E\supset F$.