AGN are a key ingredient for understanding galactic evolution. AGN-driven outflows are one of the manifestations of feedback. The AO mode for MUSE at the VLT permits to study the innermost tens of parsecs of nearby AGN in the optical. We present a detailed analysis of the ionised gas in the central regions of NGC 7130, an archetypical composite Seyfert and nuclear starburst galaxy. We achieve an angular resolution of 0.17$^{primeprime}$ (50 pc). We performed a multi-component analysis of the main ISM lines and identified nine kinematic components, six of which correspond to the outflow. The outflow is biconic and has velocities of a few $100,{rm km,s^{-1}}$ with respect to the disc. We decompose the approaching side of the outflow into a broad and a narrow component with typical velocity dispersions below and above $sim200,{rm km,s^{-1}}$, respectively. The blueshifted narrow component has substructure, in particular a collimated plume aligned with the radio jet, indicating that it may be jet-powered. The redshifted lobe is composed of two Narrow Components and a Broad Component. An additional redshifted component is seen outside the main outflow axis. Line ratio diagnostics indicate that the outflow gas in the main axis is AGN-powered whereas the off-axis component has LINER properties. The ionised gas mass outflow rate is $dot{M}=1.2pm0.7,M_{odot},{rm yr^{-1}}$ and the kinetic power is $dot{E}_{rm kin}=(2.7pm2.0)times10^{41},{rm erg,s^{-1}}$, which corresponds to $F_{rm kin}=0.12pm0.09%$ of the bolometric AGN power. The combination of high angular resolution integral field spectroscopy and a careful multi-component decomposition allows a uniquely detailed view of the outflow in NGC 7130, illustrating that AGN kinematics are more complex than traditionally derived from less sophisticated data and analyses. (abridged)
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