We say that a simple graph $G$ is Seidel integral if its Seidel spectrum consists entirely of integers. If $\alpha K_a\cup \beta K_b$ is Seidel integral we show that it belongs to one of the following classes of Seidel integral graphs % \begin {equation*} \bigg[\frac{k(2t-1)}{au}\,x_0 + \frac{m(2t-1)}{au}\,z\bigg]K_a\cup \bigg[\frac{k(2t-1)}{au}\,y_0 + \frac{\,a\,} {au}\,z\bigg](2n-1)K_b\,, \end {equation*} % where $(i)$ $a = (t + 2\ell n - (\ell+n))k + (2\ell-1)m$ and $b = (2\ell-1)m$; $(ii)$ $t,k,\ell,m,n\in \mathbb N$ such that $(m,2n-1) = 1$, $(2n-1,2t-1) =1$ and $(2\ell-1,2t-1) = 1$; $(iii)$ $\tau = (a,m(2t-1))$ such that $\tau\mid k\,(2t-1)$; $(iv)$ $(x_0,y_0)$ is a particular solution of the linear Diophantine equation $ax - m(2t-1)y = \tau$ and $(v)$ $z\ge z_0$ where $z_0$ is the least integer such that $\big(\frac{k(2t-1)}{\tau}\,x_0 + \frac{m(2t-1)}{\tau}\,z_0\big)\ge 1$ and $\big(\frac{k(2t-1)}{\tau}\,y_0 + \frac{\,a\,} {\tau}\,z_0\big)\ge 1$\,; % \begin {equation*} \bigg[\frac{2kt}{au}\,x_0 + \frac{tm}{au}\,z\bigg]K_a\cup \bigg[\frac{2kt}{au}\,y_0 + \frac{\,a\,} {au}\,z\bigg]n\, K_b\,, \end {equation*} % where $(i)$ $a = (t + \ell n)k + \ell m$ and $b = \ell m$; $(ii)$ $t,k,\ell,m,n\in \mathbb N$ such that $(m,n) = 1$, $(n,t) =1$, $(\ell,t) = 1$ and $(t + \ell n,2) = 1$; $(iii)$ $\tau = (a,tm)$ such that $\tau\mid 2kt$; $(iv)$ $(x_0,y_0)$ is a particular solution of the linear Diophantine equation $ax - (tm)y = \tau$ and $(v)$ $z\ge z_0$ where $z_0$ is the least integer such that $\big(\frac{2kt}{\tau}\, x_0 + \frac{tm}{\tau}\,z_0\big)\ge 1$ and $\big(\frac{2kt}{\tau}\, y_0 + \frac{\,a\,} {\tau}\,z\big)\ge 1$.