  This is the first in a series of papers on special Lagrangian submanifolds in
C^m. We study special Lagrangian submanifolds in C^m with large symmetry
groups, and give a number of explicit constructions. Our main results concern
special Lagrangian cones in C^m invariant under a subgroup G in SU(m)
isomorphic to U(1)^{m-2}. By writing the special Lagrangian equation as an
o.d.e. in G-orbits and solving the o.d.e., we find a large family of distinct,
G-invariant special Lagrangian cones on T^{m-1} in C^m.
  These examples are interesting as local models for singularities of special
Lagrangian submanifolds of Calabi-Yau manifolds. Such models will be needed to
understand Mirror Symmetry and the SYZ conjecture.
