Imaging of debris disks has found evidence for both eccentric and offset disks. One hypothesis is that these provide evidence for massive perturbers that sculpt the observed structures. One such disk was recently observed in the far-IR by the Herschel Space Observatory around $zeta^2$ Ret. In contrast with previously reported systems, the disk is significantly eccentric, and the system is Gyr-old. We aim to investigate the long-term evolution of eccentric structures in debris disks caused by a perturber on an eccentric orbit. Both analytical predictions and numerical N-body simulations are used to investigate the observable structures that could be produced by eccentric perturbers. The long-term evolution of the disk geometry is examined, with particular application to the $zeta^2$ Ret system. In addition, synthetic images of the disk are produced for comparison with Herschel observations. We show that an eccentric companion can produce both the observed offsets and eccentric disks. Such effects are not immediate and we characterise the timescale required for the disk to develop to an eccentric state. For the case of $zeta^2$ Ret, we place limits on the mass and orbit of the companion required to produce the observations. Synthetic images show that the pattern observed around $zeta^2$ Ret can be produced by an eccentric disk seen close to edge-on, and allow us to bring additional constraints on the disk parameters of our model (disk flux, extent). We determine that eccentric planets or stellar companions can induce long-lived eccentric structures in debris disks. Observations of such eccentric structures provide potential evidence of the presence of such a companion in a planetary system. We consider the example of $zeta^2$ Ret, whose observed eccentric disk can be explained by a distant companion at tens of AU, on an eccentric orbit ($e_pgtrsim 0.3$).
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