  In this article, we introduce $b$-semitoric systems as a generalization of
semitoric systems, specifically tailored for $b$-symplectic manifolds. The
objective of this article is to furnish a collection of examples and
investigate the distinctive characteristics of these systems. A $b$-semitoric
system is a 4-dimensional $b$-integrable system that satisfies certain
conditions: one of its momentum map components is proper and generates an
effective global $S^1$-action, and all singular points are non-degenerate and
devoid of hyperbolic components. To illustrate this concept, we provide five
examples of $b$-semitoric systems by modifying the coupled spin oscillator and
the coupled angular momenta, and we also classify their singular points.
Additionally, we describe the dynamics of these systems through the image of
their respective momentum maps.
