Exceptional collections of line bundles on the Beauville surface

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface $S$. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on $S$. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections are spires of two helices.