Some Magento extensions, such as Mirasvit Sphinx Search Ultimate, can redirect 404 pages to specific search results. But when LiteMage is on they can't achieve this redirection.
In Magento's default implementation, all 404 pages are treated as cms_index_noRoute (the full action name of the default No-route Controller). LiteMage's default configuration was designed with this in mind and will cache the output for this controller. As such, redirection won't work with LiteMage's default configuration.
Change LiteMage's default configuration so that it does not cache 404 pages from this controller.
In LiteMage's config.xml file under <default> » <litemage> » <default>, change the following from
<fullcache_routes><![CDATA[cms_index_noRoute]]></fullcache_routes>
to
<fullcache_routes></fullcache_routes>
This change is not available through Magento's Admin Panel and must be set through LiteMage's configuration file. As such, you will have to manually apply this change to LiteMage's config.xml file every time you upgrade LiteMage to a new version.