We consider a 3-dimensional Pfaffian system, whose z-component is a differential system with irregular singularity at infinity and Fuchsian at zero. In the first part of the paper, we prove that its Frobenius integrability is equivalent to the sixth Painlevé equation PVI. The coefficients of the system will be explicitly written in terms of the solutions of PVI. In this way, we remake a result of [48] and correct some misprints there. Then, we express in terms of the Stokes matrices of the irregular system the monodromy invariants p jk " TrpM j M k q of a 2-dimensional Fuchsian system with four singularities, traditionally employed in the isomonodromic deformation method for PVI. The representation of PVI as integrability condition for the above mentioned 3-dimensional Pfaffian system naturally allows to classify the branches of PVI transcendents which are holomorphic at a critical point, and such that the Pfaffian system satisfies the analyticity and semisimplicity properties described in [7]. These properties make the explicit computation possible of the monodromy data associated to the classified transcendents: we compute both the associated Stokes matrices and the invariants p jk . Finally, we compute the monodromy data parametrizing the chamber of a Dubrovin-Frobenius manifold associated with a transcendent holomorphic at x " 0.
The sixth Painlevé equation PVI is both the isomonodromy deformation condition of a 2-dimensional isomonodromic Fuchsian system and of a 3-dimensional irregular system. Only the former has been used in the literature to solve the nonlinear connection problem for PVI, through the computation of invariant quantities
. We prove a new simple formula expressing the invariants p
jk
in terms of the Stokes matrices of the irregular system, making the irregular system a concrete alternative for the nonlinear connection problem. We classify the transcendents such that the Stokes matrices and the p
jk
can be computed in terms of special functions, providing a full non-trivial class of 3-dim. examples such that the theory of non-generic isomonodromy deformations of Cotti et al (2019 Duke Math. J.
168 967–1108) applies. A sub-class of these transcendents realises the local structure of all the 3-dim Dubrovin–Frobenius manifolds with semisimple coalescence points of the type studied in Cotti et al (2020 SIGMA
16 105). We compute all the monodromy data for these manifolds (Stokes matrix, Levelt exponents and central connection matrix).
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