  We present spectroscopic follow-up of candidate luminous Ly$\alpha$ emitters
(LAEs) at $z=5.7-6.6$ in the SA22 field with VLT/X-SHOOTER. We confirm two new
luminous LAEs at $z=5.676$ (SR6) and $z=6.532$ (VR7), and also present {\it
HST} follow-up of both sources. These sources have luminosities L$_{\rm
Ly\alpha} \approx 3\times10^{43}$ erg s$^{-1}$, very high rest-frame equivalent
widths of EW$_0\gtrsim 200$ {\AA} and narrow Ly$\alpha$ lines (200-340 km
s$^{-1}$). VR7 is the most UV-luminous LAE at $z>6.5$, with M$_{1500} = -22.5$,
even brighter in the UV than CR7. Besides Ly$\alpha$, we do not detect any
other rest-frame UV lines in the spectra of SR6 and VR7, and argue that
rest-frame UV lines are easier to observe in bright galaxies with low
Ly$\alpha$ equivalent widths. We confirm that Ly$\alpha$ line-widths increase
with Ly$\alpha$ luminosity at $z=5.7$, while there are indications that
Ly$\alpha$ lines of faint LAEs become broader at $z=6.6$, potentially due to
reionisation. We find a large spread of up to 3 dex in UV luminosity for
$>L^{\star}$ LAEs, but find that the Ly$\alpha$ luminosity of the brightest
LAEs is strongly related to UV luminosity at $z=6.6$. Under basic assumptions,
we find that several LAEs at $z\approx6-7$ have Ly$\alpha$ escape fractions
$\gtrsim100$ \%, indicating bursty star-formation histories, alternative
Ly$\alpha$ production mechanisms, or dust attenuating Ly$\alpha$ emission
differently than UV emission. Finally, we present a method to compute
$\xi_{ion}$, the production efficiency of ionising photons, and find that LAEs
at $z\approx6-7$ have high values of log$_{10}(\xi_{ion}$/Hz erg$^{-1}) \approx
25.51\pm0.09$ that may alleviate the need for high Lyman-Continuum escape
fractions required for reionisation.
