Context : This paper presents the results obtained with the Multi Unit Spectroscopic Explorer ( MUSE ) for the core of the lensing cluster Abell 1689 , as part of MUSE ’ s commissioning at the ESO Very Large Telescope .

Aims : Integral-field observations with MUSE provide a unique view of the central 1 \times 1 arcmin ^ { 2 } region at intermediate spectral resolution in the visible domain , allowing us to conduct a complete census of both cluster galaxies and lensed background sources .

Methods : We performed a spectroscopic analysis of all sources found in the MUSE data cube .
Two hundred and eighty-two objects were systematically extracted from the cube based on a guided-and-manual approach .
We also tested three different tools for the automated detection and extraction of line emitters .
Cluster galaxies and lensed sources were identified based on their spectral features .
We investigated the multiple-image configuration for all known sources in the field .

Results : Previous to our survey , 28 different lensed galaxies displaying 46 multiple images were known in the MUSE field of view , most of them were detected through photometric redshifts and lensing considerations .
Of these , we spectroscopically confirm 12 images based on their emission lines , corresponding to 7 different lensed galaxies between z = 0.95 and 5.0 .
In addition , 14 new galaxies have been spectroscopically identified in this area thanks to MUSE data , with redshifts ranging between 0.8 and 6.2 .
All background sources detected within the MUSE field of view correspond to multiple-imaged systems lensed by A1689 .
Seventeen sources in total are found at z \geq 3 based on their Lyman- \alpha emission , with Lyman- \alpha luminosities ranging between 40.5 \lessapprox log ( Ly \alpha ) \lessapprox 42.5 after correction for magnification .
This sample is particularly sensitive to the slope of the luminosity function toward the faintest end .
The density of sources obtained in this survey is consistent with a steep value of \alpha \leq - 1.5 , although this result still needs further investigation .

Conclusions : These results illustrate the efficiency of MUSE in characterizing lensing clusters on one hand and in studying faint and distant populations of galaxies on the other hand .
In particular , our current survey of lensing clusters is expected to provide a unique census of sources responsible for the reionization in a representative volume at z \sim 4-7 .