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Abstract Background Movement regularities are inherently present in automated goal-directed motions of the primate's arm system. They can provide important signatures of intentional behaviours driven by sensory-motor strategies, but it remains unknown if during motor learning new regularities can be uncovered despite high variability in the temporal dynamics of the hand motions. Methods We investigated the conservation and violation of new movement regularity obtained from the hand motions traced by two untrained monkeys as they learned to reach outwardly towards spatial targets while avoiding obstacles in the dark. The regularity pertains to the transformation from postural to hand paths that aim at visual goals. Results In length-minimizing curves the area enclosed between the Euclidean straight line and the curve up to its point of maximum curvature is 1/2 of the total area. Similar trend is found if one examines the perimeter. This new movement regularity remained robust to striking changes in arm dynamics that gave rise to changes in the speed of the reach, to changes in the hand path curvature, and to changes in the arm's postural paths. The area and perimeter ratios characterizing the regularity co-varied across repeats of randomly presented targets whenever the transformation from posture to hand paths was compliant with the intended goals. To interpret this conservation and the cases in which the regularity was violated and recovered, we provide a geometric model that characterizes arm-to-hand and hand-to-arm motion paths as length minimizing curves (geodesics) in a non-Euclidean space. Whenever the transformation from one space to the other is distance-metric preserving (isometric) the two symmetric ratios co-vary. Otherwise, the symmetric ratios and their co-variation are violated. As predicted by the model we found empirical evidence for the violation of this movement regularity whenever the intended goals mismatched the actions. This was manifested in unintended curved "after-effect" trajectories executed in the absence of obstacles. In this case, the system was "perturbed" away from the symmetry but after several repeats it recovered its default state. Conclusions We propose this movement regularity as a sensory-motor transformation invariant of intentional acts.