Connection Problem for Painlevé tau Functions

<div>We derive the differential identities for isomonodromic tau functions, describing their monodromy dependence. </div><div> For Painlev\'e equations we obtain them from the relation of tau function to classical action which is a consequence of quasihomogeneity of corresponding Hamiltonians. </div><div> We use these identities to solve the connection problem for generic solution of Painlev\'e-III(D8) equation, and homogeneous Painlev\'e-II equation. </div><div> </div><div> We formulate conjectures on Hamiltonian and symplectic structure of general iso\-mo\-no\-dro\-mic deformations we obtained during our studies and check them for Painlev\'e equations.</div>