Ultrafast lasers in association to beam shaping have shown to be excellent candidates for transparent material processing [1]. Non-diffractive solutions such as Bessel beams allows for precise energy deposition since they are robust to undesired non-linear effects and as they do not distort along the propagation [2]. This offers important opportunities in laser-assisted cleaving, i.e. mechanical medium separation after single-pass laser illumination. Here we break the Bessel beam cylindrical symmetry using a novel anisotropic and non-diffractive solutions to investigate both lateral intensity contributions on material response and induced processing effect for non-cylindrical defects. Using such beam shape, we report a strong cleavability enhancement as well as an improvement of the final robustness of the separated glass in comparison with Bessel beams. We demonstrate cleaving for laser-writing speed as high as 25mm/s with ∼1μm accuracy over the whole 20mm sample length.