  In this paper the existence of analytical solutions describing $Q$-balls in a
family of deformed $O(4)$ sigma models in (1+1) dimensions has been
investigated. These models involve two complex scalar fields whose coupling
breaks the $O(4)$ symmetry group to $U(1)\times U(1)$. It has been shown that
there are two types of single $Q$-balls rotating around each of the components
of the internal space and a one-parameter family of composite $Q$-balls. These
composite solutions consist of two single $Q$-balls (separated by a distance
determined by the family parameter) spinning around each complex field with the
same internal rotation frequency.
