  The couplings between supermassive black-hole binaries and their environments
within galactic nuclei have been well studied as part of the search for
solutions to the final parsec problem. The scattering of stars by the binary or
the interaction with a circumbinary disk may efficiently drive the system to
sub-parsec separations, allowing the binary to enter a regime where the
emission of gravitational waves can drive it to merger within a Hubble time.
However, these interactions can also affect the orbital parameters of the
binary. In particular, they may drive an increase in binary eccentricity which
survives until the system's gravitational-wave signal enters the pulsar-timing
array band. Therefore, if we can measure the eccentricity from observed
signals, we can potentially deduce some of the properties of the binary
environment. To this end, we build on previous techniques to present a general
Bayesian pipeline with which we can detect and estimate the parameters of an
eccentric supermassive black-hole binary system with pulsar-timing arrays.
Additionally, we generalize the pulsar-timing array $\mathcal{F}_e$-statistic
to eccentric systems, and show that both this statistic and the Bayesian
pipeline are robust when studying circular or arbitrarily eccentric systems. We
explore how eccentricity influences the detection prospects of single
gravitational-wave sources, as well as the detection penalty incurred by
employing a circular waveform template to search for eccentric signals, and
conclude by identifying important avenues for future study.
