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We look at natural foliations on the Painlev\'e VI moduli space of regularconnections of rank 2 on $\pp ^1 -{t_1,t_2,t_3,t_4}$. These foliations arefibrations, and are interpreted in terms of the nonabelian Hodge filtration,giving a proof of the nonabelian Hodge foliation conjecture in this case. Twobasic kinds of fibrations arise: from apparent singularities, and fromquasiparabolic bundles. We show that these are transverse. Okamoto's additionalsymmetry, which may be seen as Katz's middle convolution, exchanges thequasiparabolic and apparent-singularity foliations.

Foliations on the moduli space of rank two connections on the projective line minus four points