  A previously found definition of complexity for spherically symmetric fluid
distributions [1], is extended to axially symmetric static sources. In this
case there are three different complexity factors, defined in terms of three
structure scalars obtained from the orthogonal splitting of the Riemann tensor.
All these three factors vanish, for what we consider the simplest fluid
distribution, i.e a fluid spheroid with isotropic pressure and homogeneous
energy density. However, as in the spherically symmetric case, they can also
vanish for a variety of configurations, provided the energy density
inhomogeneity terms cancel the pressure anisotropic ones in the expressions for
the complexity factors. Some exact analytical solutions of this type are found
and analyzed. At the light of the obtained results, some conclusions about the
correlation (the lack of it) between symmetry and complexity, are put forward.
