  We establish a novel procedure to analyze the entanglement properties of
extremal density matrices depending on the parameters of a finite dimensional
Hamiltonian. It was applied to a general 2-qubit Hamiltonian which could
exhibit Kramers degeneracy. This is done through the extremal density matrix
formalism, which allows to extend the conventional variational principle to
mixed states. By applying the positive partial transpose criterion in terms of
the Correlation and Schlienz-Mahler matrices on the extremal density matrices,
we demonstrate that it is possible to reach both pure and mixed entangled
states, changing properly the parameters of the Hamiltonian. For time-reversal
invariant Hamiltonians, the extremal pure states can be entangled or not and we
prove that they are not time-reversal invariants. For extremal mixed states we
have in general 5 possible cases: three of them are entangled and the other two
separable.
