Owing to the random birefringence of optical fibers, the recovery of the state of polarization (SOP) is urgently needed, especially in the nonlinear spectrum division multiplexing transmissions. Based on the variance of the polarization power ratio among symbols as the cost function, we propose a novel algorithm for the blind SOP recovery of eigenvalue communications. In the single eigenvalue transmissions with phase-shift keying or 16-ary amplitude and phase-shift keying constellations, at least 25.3 dB polarization extinction ratio can be achieved by using a block length of 30, even under 7 dB OSNR condition. It also shows that the proposed algorithm can be employed in multi-eigenvalue NFDM transmissions and full-spectrum modulated NFDM system. In the experiment, our proposed algorithm performs the same as the training symbol based method in back-to-back and less than 3000 km fiber link conditions; a maximum performance gain of 1.6 dB was obtained in ultra-long-haul condition (4300 km). It also shows that the impact of the polarization mode dispersion of a single-mode fiber on the algorithm is negligible.