Displayed in the Joomla! Plugin Manager as Search - Akeeba Ticket System.
This plugin integrates Akeeba Ticket System with Joomla!'s search feature. Please note that this is the integration with the legacy search feature, not the modern search engine-like "Smart Search" feature of Joomla!. This plugin allows search to display public tickets in its results.
No, we cannot have private tickets indexed and their results shown only to the ticket owner. This is a limitation of Joomla!'s Search, not Akeeba Ticket System.