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We present results from a search for neutrinoless double-$\beta$($0\nu\beta\beta$) decay using 36.6 g of the isotope $^{150}$Nd with datacorresponding to a live time of 5.25 y recorded with the NEMO-3 detector. Weconstruct a complete background model for this isotope, including a measurementof the two-neutrino double-$\beta$ decay half-life of $T^{2\nu}_{1/2}=$[9.34$\pm$ 0.22 (stat.) $^{+0.62}_{-0.60}$ (syst.)]$\times 10^{18}$ y for the groundstate transition, which represents the most precise result to date for thisisotope. We perform a multivariate analysis to search for \zeronu decays inorder to improve the sensitivity and, in the case of observation, disentanglethe possible underlying decay mechanisms. As no evidence for \zeronu decay isobserved, we derive lower limits on half-lives for several mechanisms involvingphysics beyond the Standard Model. The observed lower limit, assuming lightMajorana neutrino exchange mediates the decay, is $T^{0\nu}_{1/2} >$ 2.0$\times 10^{22}$ y at the 90% C.L., corresponding to an upper limit on theeffective neutrino mass of $\langle m_{\nu} \rangle$ $<$ 1.6 - 5.3 eV..

physical review. d/physical review. d.

Measurement of the 2$\nu\beta\beta$ decay half-life of $^{150}$Nd and a  search for 0$\nu\beta\beta$ decay processes with the full exposure from the  NEMO-3 detector