Let $G$ be a simple graph and let $\overline G$ denote its complement. We say that $G$ is integral if its spectrum consists of integral values. We have recently established a characterization of integral graphs which belong to the class $\overline {\alpha K_a\cup\beta K_{b,b}}$, where $mG$ denotes the $m$-fold union of the graph $G$. In this work we investigate integral graphs from the class $\overline{\alpha K_a\cup\beta K_{b,b}}$ with $\overline\lambda_1=a+b$, where $\overline\lambda_1$ is the largest eigenvalue of $\overline{\alpha K_a\cup\beta K_{b,b}}$.