A pseudocompactification

For a locally pseudocompact space $X$ let [\zeta X=X\cup cl_{\beta X}(\beta X\backslash\upsilon X).] It is proved that $\zeta X$ is the largest (with respect to the standard partial order $\le$) among all pseudocompactifications of $X$ which have compact remainder. Other characterizations of $\zeta X$ are also given.