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We present results from a $\lambda/\Delta\lambda=330$ $K$-band spectroscopic survey of $\sim$60 galaxies with $L_{IR} \geq10^{11.2} L_{\odot}$ from the IRAS Bright Galaxy Sample and its new extension to southern declinations and low Galactic latitudes. This is 61\% of all galaxies in the samples with such large luminosities, and includes 72\% (13/18) of the ultraluminous infrared bright galaxies (ULIBGs: $L_{IR} \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}} \hbox{$>$}}}10^{12} L_{\odot}$), in the observable declination range. Several key results have emerged from this first $K$-band spectroscopic survey of a statistically significant sample of luminous IRAS galaxies. (1) In general, the 2 \micron~spectra of these galaxies are dominated by features associated with stars and star formation. No active nuclei (AGN) previously unknown from optical spectroscopy were identified in our survey, despite the lower extinction at 2 \micron. If obscured AGN are present, the extinctions to the broad-line regions must be at least tens of magnitudes $A_{V}$. (2) The luminous infrared bright galaxies (LIBGs: $10^{11.2} \leq L_{IR}/L_{\odot} \leq 10^{11.96}$) have $L_{Br\gamma} \approx 10^{-5} L_{IR}$; this is consistent with extrapolations of $L_{H\alpha}/L_{IR}$ correlations and is similar to values observed in Galactic and extragalactic H{\sc ii} regions. Sufficient star formation is detected to account for the majority of the bolometric luminosity of the LIBGs. (3) $L_{K} \propto L_{IR}$ for the LIBGs, at levels consistent with those expected from the star formation inferred from (2). (4) As a group, the ULIBGs have lower $L_{Br\gamma}/L_{IR}$ ratios than the LIBGs; this may be due to extinction, since the ULIBGs also tend to have lower $L_{K}/L_{IR}$ ratios than the LIBGs. Enhanced extinction may result from high molecular gas concentrations in the ULIBGs, and might obscure their central regions, including any luminous buried AGN. (5) Powerful H$_{2}$ emission is found in virtually all sample galaxies. Both LIBGs and ULIBGs have $L_{1-0S(1)} \approx 10^{-5} L_{IR}$. H$_{2}$ line ratios indicate that the H$_{2}$ is probably shock-excited in most of the systems. (6) CO absorption significantly stronger than observed in elliptical galaxies is detected in almost all the galaxies; much of the the $K$-band continuum comes from K and M supergiants in starburst stellar populations.
