Algebraically compact Abelian groups with $\mathrm{UA}$-rings of endomorphisms

A ring $K$ is said to be a unique addition ring ($\mathrm{UA}$-ring) if on its multiplicative semigroup $(K, \cdot)$ it is possible to set only one binary operation of $+$ turning $(K, \cdot, +)$ into a ring. We call an Abelian group an $\mathrm{End}$-$\mathrm{UA}$-group if its endomorphism ring is a $\mathrm{UA}$-ring. In this paper, $\mathrm{End}$-$\mathrm{UA}$-groups are found in a class of algebraically compact Abelian groups.