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Due to the long imaging times in SPECT, patient motion is inevitable and constitutes a serious problem for any reconstruction algorithm. The measured inconsistent projection data lead to reconstruction artifacts which can significantly affect the diagnostic accuracy of SPECT if not corrected. Among the most promising attempts for addressing this cause of artifacts is the so-called data-driven motion correction methodology, implemented, for example, in the OSEM scheme. At present, this algorithm is restricted to the exclusive use of a dual-head SPECT system with perpendicular heads and incorporating in a subset only projection data obtained between a patient movement. The utilization within other SPECT systems may lead to unsatisfactory results. In this note we present a new reconstruction framework which overcomes these two shortcomings. Within the new framework, the user may choose any set of projection for the reconstruction and the scheme works for any SPECT system. As a byproduct, the well-known EM and OSEM reconstruction schemes may be written in terms of the new framework and therefore are included in the theoretical considerations. The paper is supplemented by a large set of test examples, underscoring the potential power of the proposed novel approach. Using both an academic example and images from a double-head detector we studied the extent of defects induced by simulated motion and validated the new schemes.