Control of saccadic eye movements, crucial for visual acuity during attention shifts, undergoes adaptation due to internal and external factors, such as fatigue and changing environments, critically depending on the cerebellum. Here we aim to clarify the neural mechanisms underlying saccadic adaptation in two dimensions, particularly the role of the cerebellar Purkinje cells (PC) in movement coding and plasticity. We recorded 56 PCs from the oculomotor vermis of three adult male rhesus monkeys (Macaca Mulatta) during a cross-axis adaptation task comprising three epochs. In the first pre-adaptation epoch, they made 15° saccades from a fixation point towards a visual target presented randomly at one of ten locations. Next, in the adaptation period, the target made initial horizontal jumps followed by 5° vertical leaps before the primary saccades finished. Finally, the pre-adaptation protocol repeated in the post-adaptation epoch. We analyzed the PC simple spike (SS) activity by identifying its low-dimensional structure (manifold) and its variations with the saccade angle and complex spike (CS) firing. Our preliminary findings show that, despite apparent differences, the PC-SS manifold for a saccadic eye movement has a consistent low-dimensional structure (d=4, explaining >88% variance) across movement directions, explained by a simple product of the direction-dependent multi-dimensional gain field matrix and independent latent dynamics vector. We will also present how CSs impact the directional coding of the PC-SS manifold, which can be used to probe the mechanism of the cross-axis adaptation. Our findings provide insights into the neural mechanisms for complex sensorimotor learning.