  We present a study of the galaxy Lyman-alpha luminosity function (LF) using a
sample of 17 lensing clusters observed by the MUSE/VLT. Magnification from
strong gravitational lensing by clusters of galaxies and MUSE apabilities allow
us to blindly detect LAEs without any photometric pre-selection, reaching the
faint luminosity regime. 600 lensed LAEs were selected behind these clusters in
the redshift range 2.9<$z$< 6.7, covering four orders of magnitude in
magnification-corrected Lyman-alpha luminosity (39.0<log$L$< 43.0). The method
used in this work ($V_{\text{max}}$) follows the recipes originally developed
by arXiv:1905.13696(N) (DLV19) with some improvements to better account for the
effects of lensing when computing the effective volume. The total co-moving
volume at 2.9<$z$<6.7 is $\sim$50 $10^{3}Mpc^{3}$. Our LF points in the bright
end (log L)>42 are consistent with those obtained from blank field
observations. In the faint luminosity regime, the density of sources is well
described by a steep slope, $\alpha\sim-2$ for the global redshift range. Up to
log(L)$\sim$41, the steepening of the faint end slope with redshift, suggested
by the earlier work of DLV19 is observed, but the uncertainties remain large. A
significant flattening is observed towards the faintest end, for the highest
redshift bins (log$L$<41). Using face values, the steep slope at the faint-end
causes the SFRD to dramatically increase with redshift, implying that LAEs
could play a major role in the process of cosmic reionization. The flattening
observed towards the faint end for the highest redshift bins still needs
further investigation. This turnover is similar to the one observed for the UV
LF at $z\geq6$ in lensing clusters, with the same conclusions regarding the
reliability of current results (e.g.arXiv:1803.09747(N); arXiv:2205.11526(N)).
