Equilibrium configurations are considered of a long rod constrained to lie in a plane subject to end conditions constituting a wrench. Using the Costar theory, a formulation of the problem is proposed using a reduced angular description of the director basis. On the assumption of an isotropic cross-section, it is found that flexure and torsion decouple so that the rod buckles like a planar elastic. For rods held under gravity, a condition is derived for the applied end loads required for lift-off of the localized mode under tension. For anisotropy rods, flexure and torsion are coupled and additional more complex equilibrium shapes are possible including multi-loop localized modes. Using specially adapted numerical shooting techniques such solutions, which are mathematically represented by homoclinic orbits to a periodic solution, are computed, and conditions for lift-off of the single-loop solutions are calculated as a function of the applied loads and an anisotropy parameter.
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