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We present a catalogue of 261 new infrared selected members of the 3C129 galaxy cluster. The cluster, located at $z \approx$ 0.02, forms part of the Perseus-Pisces filament and is obscured at optical wavelengths due to its location in the zone of avoidance. We identified these galaxies using the $J-$ and $K-$band imaging data provided by the UKIDSS Galactic Plane Survey within an area with a radius of $1.1^{\circ}$ centred on the X-ray emission of the cluster at $\ell, b \approx 160.52^{\circ}, 0.27^{\circ}$. A total of 26 of the identified galaxy members have known redshifts 24 of which are from our 2016 Westerbork HI survey and two are from optical spectroscopy. An analysis of the galaxy density at the core of the 3C129 cluster shows it to be less dense than the Coma and Norma clusters, but comparable to the galaxy density in the core of the Perseus cluster. From an assessment of the spatial and velocity distributions of the 3C129 cluster galaxies that have redshifts, we derived a velocity of $cz = 5227 \pm 171$ km/s and $\sigma = 1097 \pm 252$ km/s for the main cluster, with a substructure in the cluster outskirts at $cz = 6923 \pm 71$ km/s with $\sigma = 422 \pm 100$ km/s. The presence of this substructure is consistent with previous claims based on the X-ray analysis that the cluster is not yet virialised and may have undergone a recent merger., Comment: 33 pages, 16 figures, includes an appendix, accepted for publication in A&A