  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.
