  We consider limits of $\mathcal{N}=4$ super Yang-Mills (SYM) theory that
approach BPS bounds and for which an $SU(1,1)$ structure is preserved. The
resulting near-BPS theories become non-relativistic, with a $U(1)$ symmetry
emerging in the limit that implies the conservation of particle number. They
are obtained by reducing $\mathcal{N}=4$ SYM on a three-sphere and subsequently
integrating out fields that become non-dynamical as the bounds are approached.
Upon quantization, and taking into account normal-ordering, they are consistent
with taking the appropriate limits of the dilatation operator directly, thereby
corresponding to Spin Matrix theories, found previously in the literature. In
the particular case of the $SU(1,1|1)$ near-BPS/Spin Matrix theory, we find a
superfield formulation that applies to the full interacting theory. Moreover,
for all the theories we find tantalizingly simple semi-local formulations as
theories living on a circle. Finally, we find positive-definite expressions for
the interactions in the classical limit for all the theories, which can be used
to explore their strong coupling limits. This paper will have a companion paper
in which we explore BPS bounds for which a $SU(2,1)$ structure is preserved.
