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The unitary group acting on the Hilbert space $${\cal H}:=(C^2)^{\otimes 3}$$ of three quantum bits admits a Lie subgroup, $$U^{S_3}(8)$$, of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space $${\cal H}$$ splits into three invariant subspaces of dimensions $$4$$, $$2$$ and $$2$$ respectively, each corresponding to an irreducible representation of $su(2)$. The subspace of dimension $$4$$ is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under $$U^{S_3}(8)$. This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup $$U^{S_3}(8)$$. As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin $$\frac{1}{2}$$ particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement. 

We propose a method to analyze the three-dimensional nonholonomic system known as the Brockett integrator and to derive the (energy) optimal trajectories between two given points. For systems with nonholonomic constraint, it is well-known that the energy optimal trajectories corresponds to sub-Riemannian geodesics under a proper sub-Riemannian metric. Our method uses symmetry reduction and an analysis of the quotient space associated with the action of a (symmetry) group on R^3. By lifting the Riemannian geodesics with respect to an appropriate metric from the quotient space back to the original space R^3, we derive the optimal trajectories of the original problem.