Abstract
The symmetry energy is a general property of all two component fermionic systems. In nuclear matter the problem is to get direct experimental information on the isovector part of the in medium nuclear interaction and then on the density dependence of the symmetry energy. In this context it is important to select the most sensitive observables.

The only way to reach regions away from saturation is to study the dynamical response in Heavy Ion Collisions with exotic nuclear beams. Here this is obtained in a transport theory based on a microscopic stochastic mean field approach with effective interactions. New reaction mechanisms, rather sensitive to the density dependence of the symmetry energy, are also presented. The comparison is made with already existing data and suggestions are given for new experiments [1-3]. Connections to neutron star properties are rapidly commented. This is the detailed plan of the report:

1. Region around saturation. The slope parameter L is discussed with expected effects on nuclear compressibility and extension of the neutron skin. In particular the sensitivity of the Pigmy Dipole Resonance in stressed.
2. Low density region. The new Dynamical Dipole Mode for the charge equilibration in fusion dynamics is suggested as a good observable. Moreover the competition Fusion vs Break-up appears interesting . In Multifragmentation events at the Fermi energies we expect to see signals of an Isospin Distillation. The new dissipative Neck-Fragmentation process in more peripheral events presents good evidence of an isospin transport from Projectile-like and Target-like partners to the fragments, also sensitive to the symmetry energy slope below saturation.
3. High density region reached at intermediate energies. Here the particle emission and flows are analyzed. In particular the elliptic flows (squeeze-out) for n/p and for light charge asymmetric ions are suggested. Now even the poorly known n/p mass splitting in asymmetric matter is expected to give noticeable contributions. Nucleon resonances and mesons are also produced with some interesting isospin effects, in particular around the thresholds. High transverse momentum selections are requested in order to observe particles produced in high density regions.
4. Transition to quark matter at high baryon densities. The contribution of the nuclear symmetry energy can be very important for the transition region, expected at lower densities for isospin asymmetric matter and with an isospin distillation effect for the quark clusters formed in the mixed phase. Unfortunately our poor knowledge of the QCD equation of state at high chemical potentials is strongly limiting the reliability of the results. In any case observables can be suggested for experiments in the ten GeV/A region with neutron-rich ions.

[1] V.Baran, M.Colonna, V.Greco, M.Di Toro, Phys. Rep. 410, 335 (2005).
[2] M.Di Toro, V.Baran, M.Colonna, V.Greco, J.Phys.G Nucl.Part.Phys 37, 083101 (2010), Review paper
[3] M.Colonna, V.Baran, M.Di Toro, Eur.Phys.J. A50, 30 (2014), Topical Issue