Real Liouvillian Extensions of Partial Differential Fields

In this paper, we establish Galois theory for partial differential systems defined over formally real fields with a closed field of constants and $p$-adic $p$-adically constants. For an integrable system such field, prove that there exists (resp. $p$-adic) Picard-Vessiot extension. Moreover, obtain uniqueness result We give adequate definition the group fundamental theorem in setting. apply obtained correspondence to characterise Liouvillian extensions by means split solvable linear algebraic groups. present some examples dynamical indicate possibilities further development methods systems.