  This paper is about the relation of the geometry of Lie groupoids over a
fixed compact manifold and the geometry of their (infinite-dimensional)
bisection Lie groups. In the first part of the paper we investigate the
relation of the bisections to a given Lie groupoid, where the second part is
about the construction of Lie groupoids from candidates for their bisection Lie
groups. The procedure of this second part becomes feasible due to some recent
progress in the infinite-dimensional Frobenius theorem, which we heavily
exploit. The main application to the prequantisation of (pre)symplectic
manifolds comes from an integrability constraint of closed Lie subalgebras to
closed Lie subgroups. We characterise this constraint in terms of a modified
discreteness conditions on the periods of that manifold.
