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The satellite Linus orbiting the main-belt asteroid (22) Kalliope exhibitedoccultation and transit events in late 2021. A photometric campaign wasorganized and observations were taken by the TRAPPIST-South, SPECULOOS-Artemis,OWL-Net, and BOAO telescopes, with the goal to constrain models of this system.Our dynamical model is complex, with multipoles (up to the order $\ell = 2$),internal tides, and external tides. The model was constrained by astrometry(spanning 2001--2021), occultations, adaptive-optics imaging, calibratedphotometry, as well as relative photometry. Our photometric model wassubstantially improved. A new precise (${<}\,0.1\,{\rm mmag}$) light curvealgorithm was implemented, based on polygon intersections, which are computedexactly -- by including partial eclipses and partial visibility of polygons.Moreover, we implemented a `cliptracing' algorithm, based again on polygonintersections, in which partial contributions to individual pixels are computedexactly. Both synthetic light curves and synthetic images are then very smooth.Based on our combined solution, we confirmed the size of Linus, $(28\pm1)\,{\rm km}$. However, this solution exhibits some tension between the lightcurves and the PISCO speckle-interferometry dataset. In most solutions, Linusis darker than Kalliope, with the albedos $A_{\rm w} = 0.40$ vs. $0.44$. Thisis confirmed on deconvolved images. A~detailed revision of astrometric dataallowed us to revise also the $J_2 \equiv -C_{20}$ value of Kalliope. Mostimportantly, a~homogeneous body is excluded. For a differentiated body, twosolutions exist: low-oblateness ($C_{20} \simeq -0.12$), with a~spherical ironcore, and alternatively, high-oblateness ($C_{20} \simeq -0.22$) with anelongated iron core. These correspond to the low- and high-energy collisions,respectively, studied by means of SPH simulations in our previous work.

2021 occultations and transits of Linus orbiting (22) Kalliope: I. Polygonal and `cliptracing' algorithm