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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 newluminous LAEs at $z=5.676$ (SR6) and $z=6.532$ (VR7), and also present {\itHST} follow-up of both sources. These sources have luminosities L$_{\rmLy\alpha} \approx 3\times10^{43}$ erg s$^{-1}$, very high rest-frame equivalentwidths of EW$_0\gtrsim 200$ {\AA} and narrow Ly$\alpha$ lines (200-340 kms$^{-1}$). 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 anyother rest-frame UV lines in the spectra of SR6 and VR7, and argue thatrest-frame UV lines are easier to observe in bright galaxies with lowLy$\alpha$ equivalent widths. We confirm that Ly$\alpha$ line-widths increasewith Ly$\alpha$ luminosity at $z=5.7$, while there are indications thatLy$\alpha$ lines of faint LAEs become broader at $z=6.6$, potentially due toreionisation. 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 brightestLAEs 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, alternativeLy$\alpha$ production mechanisms, or dust attenuating Ly$\alpha$ emissiondifferently than UV emission. Finally, we present a method to compute$\xi_{ion}$, the production efficiency of ionising photons, and find that LAEsat $z\approx6-7$ have high values of log$_{10}(\xi_{ion}$/Hz erg$^{-1}) \approx25.51\pm0.09$ that may alleviate the need for high Lyman-Continuum escapefractions required for reionisation.

Spectroscopic properties of luminous Lyman-{\alpha} emitters at $z \approx 6 - 7$ and comparison to the Lyman-break population