  The symmetry operators for the Laplacian in flat space were recently
described and here we consider the same question for the square of the
Laplacian. Again, there is a close connection with conformal geometry. There
are three main steps in our construction. The first is to show that the symbol
of a symmetry is constrained by an overdetermined partial differential
equation. The second is to show existence of symmetries with specified symbol
(using a simple version of the AdS/CFT correspondence). The third is to compute
the composition of two first order symmetry operators and hence determine the
structure of the symmetry algebra. There are some interesting differences as
compared to the corresponding results for the Laplacian.
