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We consider a 3-dimensional Pfaffian system, whose z-component is adifferential system with irregular singularity at infinity and Fuchsian atzero. In the first part of the paper, we prove that its Frobenius integrabilityis equivalent to the sixth Painlev\'e equation PVI. The coefficients of thesystem will be explicitly written in terms of the solutions of PVI. In thisway, we remake a result of [44, 61]. We then express in terms of the Stokesmatrices of the 3x3 irregular system the monodromy invariants p_{jk}=Tr(M_jM_k)of the 2-dimensional isomonodromic Fuchsian system with four singularities,traditionally associated to PVI [23, 55] and used to solve the non-linearconnection problem. Several years after [44, 61], the authors of [14] showedthat the computation of the monodromy data of a class of irregular systems maybe facilitated in case of coalescing eigenvalues. This coalescence correspondsto the critical points (fixed singularities) of PVI. In the second part of thepaper, we classify the branches of PVI transcendents holomorphic at a criticalpoint such that the analyticity and semisimplicity properties described in [14]are satisfied, and we compute the associated Stokes matrices and the invariantsp_{jk}. Finally, we compute the monodromy data parametrizing the chamber of a3-dim Dubrovin-Frobenius manifold associated with a transcendent holomorphic atx=0.

The sixth Painleve' equation as isomonodromy deformation of an irregular  system: monodromy data, coalescing eigenvalues, locally holomorphic  transcendents and Frobenius manifolds