  Real filaments are not perfectly homogeneous. Most of them have various
materials composition and shapes making their stiffnesses not constant along
the arclength. We investigate the existence of circular and helical equilibrium
solutions of an intrinsically straight rod with varying bending and twisting
stiffnesses, within the framework of the Kirchhoff model. The planar ring
equilibrium solution only exists for a rod with a given form of variation of
the bending stiffness. We show that the well known circular helix is not an
equilibrium solution of the static Kirchhoff equations for a rod with non
constant bending stiffness. Our results may provide an explanation for the
variation of the curvature seen in small closed DNAs immersed in a solution
containing Zn^{2+}, and in the DNA wrapped around a nucleosome.
