  For an oriented surface link $S$, we can take a satellite construction called
a 2-dimensional braid over $S$, which is a surface link in the form of a
covering over $S$. We demonstrate that 2-dimensional braids over surface links
are useful for showing the distinctness of surface links. We investigate
non-trivial examples of surface links with free abelian group of rank two,
concluding that their link types are infinitely many.
