Abstract: An integrable Poisson manifold is said to be of strong compact type if the source 1-connected groupoid integrating it is compact. A trivial class of such manifolds is that of compact symplectic manifolds with finite fundamental group, but beyond that finding examples is difficult. The first non-trivial example was given by D. Martínez Torres in 2014. The construction there is inspired by D. Kotschick’s construction of a free symplectic circle action with contractible orbits. In this talk I will go over the original construction, recalling the relevant results on Poisson manifolds of compact types as well as the geometry of the moduli spaces of K3 surfaces, and then modify the construction to obtain more examples. In the end, we will have for every strongly integral affine 2-torus (i.e. integral affine 2-torus with integral translational part) a Poisson manifold of strong compact type having said torus as its leaf space.