  A major current goal of noncommutative geometry is the classification of
noncommutative projective surfaces. The generic case is to understand algebras
birational to the Sklyanin algebra. In this work we complete a considerable
component of this problem. Let S denote the 3-dimensional Sklyanin algebra over
an algebraically closed field, and assume that S is not a finite module over
its centre. In earlier work Rogalski, Sierra and Stafford classified the
maximal orders inside the 3-Veronese of S. We complete and extend their work
and classify all maximal orders inside S. As in Rogalski, Sierra and Stafford's
work, these can be viewed as blowups at (possibly non-effective) divisors. A
consequence of this classification is that maximal orders are automatically
noetherian among other desirable properties.
