A non-abelian version of Deligne's Fixed Part Theorem

We formulate and prove a non-abelian analog of Deligne’s Fixed Part theorem on Hodge classes, revisiting previous work of Jost–Zuo, Katzarkov–Pantev and Landesman–Litt. To this aim we study algebraically isomonodromic extensions of local systems and we relate them to variations of Hodge structures, for example we show that the Mumford-Tate group at a generic point stays constant in an algebraically isomonodromic extension of a variation of Hodge structure. v2: a few typos ironed and Thm 1.1 5) completed. v3: there was a Schlamassel leading to a mix-up of files. Apologies. Else identical version (one minor change). v5 final version. Appears in Alg. Geom.