Recently, pear-shaped nuclei have been discovered. This discovery has interesting implications, as noted in this question: Do pear-shaped nuclei really have anything to do with time travel?
At the moment, the nuclei known to be pear-shaped are 220Rn (half-life of 55.6 s), 224Rn (half-life of 3.63 days)  and 144Ba (half-life of 11.5 s) .
Why are all pear-shaped nuclei so short lived? Is there any reason preventing pear-shaped nuclei to be stable?
 L. P. Gaffney, P. A. Butler et al.:Studies of pear-shaped nuclei using accelerated radioactive beams. Nature 497, 199–204 (09 May 2013) (E-print on L.P. Gaffney's L.U. page)
 Bucher, B. et al.: Direct Evidence of Octupole Deformation in Neutron-Rich 144Ba. Phys. Rev. Lett. 116, 112503 (2016) (link to arXiv preprint)