  We have performed NMR experiments on the quasi one-dimensional frustrated
spin-1/2 system LiCuVO$_4$ in magnetic fields $H$ applied along the c-axis up
to field values near the saturation field $H_{\rm sat}$. For the field range
$H_{\rm c2}<H<H_{\rm c3}$ ($\mu_0H_{\rm c2}\approx 7.5$T and $\mu_0H_{\rm c3} =
[40.5 \pm 0.2]$T) the $^{51}$V NMR spectra at $T$ = 380mK exhibit a
characteristic double-horn pattern, as expected for a spin-modulated phase in
which the magnetic moments of Cu$^{2+}$ ions are aligned parallel to the
applied field $H$ and their magnitudes change sinusoidally along the magnetic
chains. For higher fields, the $^{51}$V NMR spectral shape changes from the
double-horn pattern into a single Lorentzian line. For this Lorentzian line,
the internal field at the $^{51}$V nuclei stays constant for $\mu_0 H > 41.4$T,
indicating that the majority of magnetic moments in LiCuVO$_4$ are already
saturated in this field range. This result is inconsistent with the previously
observed linear field dependence of the magnetization $M(H)$ for $H_{\rm
c3}<H<H_{\rm sat}$ with $\mu_0H_{\rm sat}=45$T [L. E. Svistov {\it et al}.,
JETP Letters {\bf 93}, 21 (2011)]. We argue that the discrepancy is due to
non-magnetic defects in the samples. The results of the spin-lattice relaxation
rate of $^7$Li nuclei indicate an energy gap which grows with field twice as
fast as the Zeeman energy of a single spin, therefore, suggesting that the
two-magnon bound state is the lowest energy excitation. The energy gap tends to
close at $\mu_0H \approx 41$T. Our results suggest that the theoretically
predicted spin-nematic phase, if it exists in LiCuVO$_4$, can be established
only within the narrow field range $40.5 < \mu_0 H < 41.4$T .
