HCCNC and HNC₃ are less commonly found isomers of cyanoacetylene, HC₃N, a molecule that is widely found in diverse astronomical sources. We want to know if HNC₃ is present in sources other than the dark cloud TMC-1 and how its abundance is relative to that of related molecules. We used the Astrochemical Studies At IRAM unbiased spectral survey at IRAM 30 m towards the prototypical pre-stellar core L1544 to search for HNC₃ and HCCNC which are by-product of the HC₃NH⁺ recombination, previously detected in this source. We performed a combined analysis of published HNC₃ microwave rest frequencies with thus far unpublished millimetre data because of issues with available rest frequency predictions. We determined new spectroscopic parameters for HNC₃, produced new predictions and detected it towards L1544. We used a gas-grain chemical modelling to predict the abundances of N-species and compare with the observations. The modelled abundances are consistent with the observations, considering a late stage of the evolution of the pre-stellar core. However, the calculated abundance of HNC₃ was found 5-10 times higher than the observed one. The HC₃N, HNC₃, and HCCNC versus HC₃NH⁺ ratios are compared in the TMC-1 dark cloud and the L1544 pre-stellar core.