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The Hamiltonian approach to isomonodromic deformation systems for genericrational covariant derivative operators on the Riemann sphere, having anymatrix dimension $r$ and any number of isolated singularities of arbitraryPoincar\'e rank, is derived using the split classical rational $R$-matrixPoisson bracket structure on the dual space $L^*{\frak gl}(r)$ of the loopalgebra $L{\frak gl}(r)$. Nonautonomous isomonodromic counterparts ofisospectral systems are obtained by identifying the deformation parameters asCasimir elements on the phase space. These are shown to coincide with thehigher Birkhoff invariants determining the local asymptotics near to irregularsingular points, together with the pole loci. They appear as the negative powercoefficients in the principal part of the Laurent expansion of the fundamentalmeromorphic differential on the associated spectral curve, while thecorresponding dual spectral invariant Hamiltonians appear as the "mirror image"positive power terms in the analytic part. Infinitesimal isomonodromicdeformations are generated by the sum of the Hamiltonian vector field and anexplicit derivative vector field that is transversal to the symplecticfoliation.

Hamiltonian structure of isomonodromic deformation dynamics in linear systems of PDE's