Context. Multi-phase outflows play a central role in galaxy evolution shaping the properties of galaxies. Understanding outflows and their effects in low luminosity active galactic nuclei (AGNs), such as low ionisation nuclear emission line regions (LINERs), is essential. LINERs bridge the gap between normal and active galaxies, being the most numerous AGN population in the local Universe. Aims. Our goal is to analyse the kinematics and ionisation mechanisms of the multi-phase gas of NGC 1052, the prototypical LINER, in order to detect and map the ionised and neutral phases of the putative outflow. Methods. We obtained Very Large Telescope MUSE and Gran Telescopio Canarias MEGARA optical integral field spectroscopy data for NGC 1052. In addition to stellar kinematics maps, by modelling spectral lines with multiple Gaussian components, we obtained flux, kinematic, and excitation maps of both ionised and neutral gas. Results. The stars are distributed in a dynamically hot disc (V/σ ∼ 1.2), with a centrally peaked velocity dispersion map (σc = 201 ± 10 km s−1) and large observed velocity amplitudes (ΔV = 167 ± 19 km s−1). The ionised gas, probed by the primary component is detected up to ∼30″ (∼3.3 kpc) mostly in the polar direction with blue and red velocities (∣V∣ < 250 km s−1). The velocity dispersion map shows a notable enhancement (σ > 90 km s−1) crossing the galaxy along the major axis of rotation in the central 10″. The secondary component has a bipolar morphology, velocity dispersion larger than 150 km s−1, and velocities up to 660 km s−1. A third component is detected with MUSE (and barely with MEGARA), but it is not spatially resolved. The broad-line region (BLR) component (used to model the broad Hα emission only) has a full width at half maximum of 2427 ± 332 and 2350 ± 470 km s−1 for MUSE and MEGARA data, respectively. The maps of the NaD absorption indicate optically thick neutral gas with complex kinematics. The velocity field is consistent with a slow rotating disc (ΔV = 77 ± 12 km s−1), but the velocity dispersion map is off-centred without any counterpart in the (centrally peaked) flux map. Conclusions. We found evidence of an ionised gas outflow (secondary component) with a mass of 1.6 ± 0.6 × 105 M⊙, and mass rate of 0.4 ± 0.2 M⊙ yr−1. The outflow is propagating in a cocoon of gas with enhanced turbulence and might be triggering the onset of kiloparsec-scale buoyant bubbles (polar emission), both probed by the primary component. Taking into account the energy and kinetic power of the outflow (1.3 ± 0.9 × 1053 erg and 8.8 ± 3.5 × 1040 erg s−1, respectively) as well as its alignment with both the jet and the cocoon, and that the gas is collisionally ionised (due to gas compression), we consider that the most likely power source of the outflow is the jet, although some contribution from the AGN is possible. The hints of the presence of a neutral gas outflow are weak.
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