  The space of m-ary differential operators acting on weighted densities is a
(m+1)-parameter family of modules over the Lie algebra of vector fields. For
almost all the parameters, we construct a canonical isomorphism between this
space and the corresponding space of symbols as sl(2)-modules. This yields to
the notion of the sl(2)-equivariant symbol calculus for m-ary differential
operators. We show, however, that these two modules cannot be isomorphic as
sl(2)-modules for some particular values of the parameters. Furthermore, we use
the symbol map to show that all modules of second-order m-ary differential
operators are isomorphic to each other, except for few modules called singular.
