The eigenbeam-ESPRIT (EB-ESPRIT) is well known as a high-resolution parametric direction-of-arrival (DOA) estimation technique for a spherical microphone array. Unlike other eigenbeam beamformers such as EB-MVDR and EB-MUSIC, there is no need for exhaustive grid-search with EB-ESPRIT. However, when sources are positioned near the equator, the EB-ESPRIT inevitably induces a singularity problem due to the singularity of its tangent function utilized as a directional parameter. Here, a new EB-ESPRIT technique based on a nonsingular directional parameter is proposed. The proposed technique is derived from two sine-based recurrence relations, and hence, can fundamentally avoid the singularity problems and identify more sources than the conventional EB-ESPRIT.