The goal of this article is to present the algorithm for DMLC leaf control capable of delivering IMRT to tumors that experience motion in two dimensions in the beams eye view (BEV) plane. The generic, two-dimensional (2D) motion of the projection of the rigid target on BEV plane can be divided into two components. The first component describes the motion of the projection of the target along the x axis (parallel to the MLC leaf motions) and the other describes the motion of the target projection on the y axis (perpendicular to the leaf motion direction). First, time optimal leaf trajectories are calculated independently for each leaf pair of the MLC assembly to compensate the x-axis component of the 2D motion of the target on the BEV. These leaf trajectories are then synchronized following the mid time (MT) synchronization procedure. To compensate for the y-axis component of the motion of the target projection on the BEV plane, the procedure of "switching" leaf pair trajectories in the upward (or downward) direction is executed when the target's BEV projection moves upward (or downward) from its equilibrium position along the y axis. When the intensity function is a 2D histogram, the error between the intended and delivered intensity in 2D DMLC IMRT delivery will depend on the shape of the intensity map and on the MLC physical constraint (leaf width and maximum admissible leaf speed). The MT synchronization of leaf trajectories decreases the impact of above constraints on the error in 2D DMLC IMRT intensity map delivery. The proof is provided, that if hardware constraints in the 2D DMLC IMRT delivery strategy are removed, the errors between planned and delivered 2D intensity maps are entirely eliminated. Examples of 2D DMLC IMRT delivery to rigid targets moving along elliptical orbits on BEV planes are calculated and analyzed for 20 clinical fluence maps. The comparisons between the intensity delivered without motion correction, with motion correction along x axis only, and with motion correction for full 2D motion of the target are calculated and quantitatively evaluated. The fluence maps were normalized to 100 MU and the rms difference between the desired and delivered fluence was 12 MU for no motion compensation, 11.18 MU for 1D compensation, and 4.73 MU for 2D motion compensations. The advantage of correcting for full 2D motion of target projected on the BEV plane is demonstrated.