On parameterized differential Galois extensions

We prove some existence results on parameterized strongly normal extensions for logarithmic equations. We generalize a result in Wibmer (2012) [16]. We also consider an extension of the results in Kamensky and Pillay (2014) [4] from the ODE case to the parameterized PDE case. More precisely, we show that if DD and Δ are two distinguished sets of derivations and (KD,Δ)(KD,Δ) is existentially closed in (K,Δ)(K,Δ), where K   is a D∪ΔD∪Δ-field of characteristic zero, then every (parameterized) logarithmic equation over K has a parameterized strongly normal extension.