  Motivated by M\"obius transformation for symmetrical points under the
generalized circle in complex plane, the system of symmetrical spin coherent
states corresponding to antipodal qubit states is introduced. It implies the
maximally entangled spin coherent states basis, which in the limiting cases
reduces to the Bell basis. A specific property of our symmetric image coherent
states is that they never become unentangled for any value of $\psi$ from
complex plane. By the reduced density matrix and the concurrence determinant
methods, it is shown that our basis is maximally entangled. In addition we find
that the average of spin operators in these states vanish, as it must be
according to another, operational definition of completely entangled states.
Universal one qubit and two qubit gates in this new basis are calculated and
time evolution of these states for some spin systems is derived. We find that
the average energy for XYZ model in two qubit case (Q symbol of H) shows
regular finite energy localized structure with characteristic extremum points,
and appears as a soliton in maximally entangled two qubit phase space.
Generalizations to three and higher qubit states are discussed.
