  Heavy Ion Collisions (HIC) represent a unique tool to probe the in-medium
nuclear interaction in regions away from saturation. In this report we present
a selection of new reaction observables in dissipative collisions particularly
sensitive to the symmetry term of the nuclear Equation of State ($Iso-EoS$). We
will first discuss the Isospin Equilibration Dynamics. At low energies this
manifests via the recently observed Dynamical Dipole Radiation, due to a
collective neutron-proton oscillation with the symmetry term acting as a
restoring force. At higher beam energies Iso-EoS effects will be seen in an
Isospin Diffusion mechanism, via Imbalance Ratio Measurements, in particular
from correlations to the total kinetic energy loss. For fragmentation reactions
in central events we suggest to look at the coupling between isospin
distillation and radial flow. In Neck Fragmentation reactions important Iso-EoS
information can be obtained from fragment isospin content, velocity and
alignement correlations. The high density symmetry term can be probed from
isospin effects on heavy ion reactions at relativistic energies (few AGeV
range), in particular for high transverse momentum selections of the reaction
products. Rather isospin sensitive observables are proposed from
nucleon/cluster emissions, collective flows and meson production. The
possibility to shed light on the controversial neutron/proton effective mass
splitting in asymmetric matter is also suggested.
  A large symmetry repulsion at high baryon density will also lead to an
"earlier" hadron-deconfinement transition in n-rich matter. The binodal
transition line of the (T,\rho_B) diagram is lowered to a region accessible
through heavy ion collisions in the energy range of the new planned facilities,
e.g. the FAIR/NICA projects. Some observable effects of the formation of a
Mixed Phase are suggested, in particular a Neutron Trapping mechanism. The
dependence of the results on a suitable treatment of the isovector part of the
interaction in effective QCD Lagrangian approaches is critically discussed. We
stress the interest of this study in nuclear astrophysics, in particular for
supernovae explosions and neutron star structure, where the knowledge of the
Iso-EoS is important at low as well as at high baryon density.
