  This paper considers the local integrability condition for generalised
translation-invariant systems and its relation to the Calder\'on integrability
condition, the temperateness condition and the uniform counting estimate. It is
shown that sufficient and necessary conditions for satisfying the local
integrability condition are closely related to lower and upper bounds on the
number of lattice points that intersect with the translates of a compact set.
The results are complemented by examples that illustrate the crucial interplay
between the translation subgroups and the generating functions of the system.
