We present a geometric setting for the differential Galois theory of G-invariant connections with parameters. As an application of some classical results on differential algebraic groups and Lie algebra bundles, we see that the Galois group of a connection with parameters with simple structural group G is determined by its isomonodromic deformations. This allows us to compute the Galois groups with parameters of the general Fuchsian special linear system and of Gauss hypergeometric equation.

We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie's finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential invariants, and we expose a general method that allow to compute differential invariants from joint invariants.

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The Malgrange–Galois groupoid of Painlevé IV equations is known to be, for very general values of parameters, the pseudogroup of transformations of the phase space preserving a volume form, a time form and the equation. Here we compute the Malgrange–Galois groupoid of the Painlevé VI family including all parameters as new dependent variables. We conclude that it is the pseudogroup of transformations preserving the parameter values, the differential of the independent variable, a volume form in the dependent variables and the equation. This implies that a solution of a Painlevé VI equation depending analytically on the parameters does not satisfy any new partial differential equation (including derivatives with respect to parameters) which is not derived from Painlevé VI.

The Malgrange-Galois groupoid of Painlevé IV equations is known to be, for very general values of parameters, the pseudogroup of transformations of the phase space preserving a volume form, a time form and the equation. Here we compute the Malgrange-Galois groupoid of Painlevé VI family including all parameters as new dependent variables. We conclude it is the pseoudogroup of transformations preserving parameter values, the differential of the independent variable, a volume form in the dependent variables and the equation. This implies that a solution of Painlevé VI depending analytically on parameters does not satisfy any new partial differential equation (including derivatives w. r. t. parameters) which is not derived from Painlevé VI.