Geodesic
The spectrum and separability of mixed two-qubit $X$-states
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 270-285

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The separable mixed two-qubit $X$-states are classified in accordance with the degeneracies in the spectrum of density matrices. It is shown that there are four classes of separable $X$-states, among them: one four-dimensional family, a pair of two-dimensional families, and a single zero-dimensional maximally mixed state. 
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     author = {A. Khvedelidze and A. Torosyan},
     title = {The spectrum and separability of mixed two-qubit $X$-states},
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A. Khvedelidze; A. Torosyan. The spectrum and separability of mixed two-qubit $X$-states. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 270-285. http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a18/