We present an analysis of HI Lyman-alpha emission in deep VLT/MUSE observations of two highly magnified and extended galaxies at z=3.5 and 4.03, including a newly discovered, almost complete Einstein ring. While these Lyman-alpha haloes are intrinsically similar to the ones typically seen in other MUSE deep fields, the benefits of gravitational lensing allows us to construct exceptionally detailed maps of Lyman-alpha line properties at sub-kpc scales. By combining all multiple images, we are able to observe complex structures in the Lyman-alpha emission and uncover small (~ 120 km/s in Lyman-alpha peak shift), but significant at > 4 sigma, systematic variations in the shape of the Lyman-alpha line profile within each halo. Indeed, we observe a global trend for the line peak shift to become redder at large radii, together with a strong correlation between the peak wavelength and line width. This systematic intrahalo variation is markedly similar to the object-to-object variations obtained from the integrated properties of recent large samples. Regions of high surface brightness correspond to relatively small line shifts, which could indicate that Lyman-alpha emission escapes preferentially from regions where the line profile has been less severely affected by scattering of Lyman-alpha photons.
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