  About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by
means of five invariants. Standard examples are the coupled spin oscillator on
$\mathbb{S}^2 \times \mathbb{R}^2$ and coupled angular momenta on $\mathbb{S}^2
\times \mathbb{S}^2$, both having exactly one focus-focus singularity. But so
far there were no explicit examples of systems with more than one focus-focus
singularity which are semitoric in the sense of that classification. This paper
introduces a 6-parameter family of integrable systems on $\mathbb{S}^2 \times
\mathbb{S}^2$ and proves that, for certain ranges of the parameters, it is a
compact semitoric system with precisely two focus-focus singularities. Since
the twisting index (one of the semitoric invariants) is related to the
relationship between different focus-focus points, this paper provides systems
for the future study of the twisting index.
