In this paper we prove a convexity and fibre-connectedness theorem for proper maps constructed by Thimms trick on a connected Hamiltonian $G$-space $M$ that generate a Hamiltonian torus action on an open dense submanifold. Since these maps only generate a Hamiltonian torus action on an open dense submanifold of $M$, convexity and fibre-connectedness do not follow immediately from Atiyah-Guillemin-Sternbergs convexity theorem, even if $M$ is compact. The core contribution of this paper is to provide a simple argument circumventing this difficulty. In the case where the map is constructed from a chain of subalgebras we prove that the image is given by a list of inequalities that can be computed explicitly. This generalizes the famous example of Gelfand-Zeitlin systems on coadjoint orbits introduced by Guillemin and Sternberg. Moreover, we prove that if such a map generates a completely integrable torus action on an open dense submanifold of $M$, then all its fibres are smooth embedded submanifolds.
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