Bounding a domain that contains all compact invariant sets of the bloch system

In this paper we consider the localization problem of compact invariant sets of the Bloch system describing dynamics of an ensemble of spins in an external magnetic field. Our main results are related to finding a domain containing all compact invariant sets of the Bloch system. This domain is described as an intersection of one-parameter set of balls with two half spaces. Further, we describe the location of periodic orbits respecting two circular paraboloids and one semipermeable plane. In addition, we find conditions under which the origin is the unique compact invariant set. Finally, taking the Bloch system in cylindrical coordinates we construct one first integral for some specific restriction imposed on its parameters and, we also establish conditions under which this system has no compact invariant sets.