there isn't anything parallel to RunPublicQuery in the rest api today. this is needed for moving the GUI to work over REST API for things like getting the list of domains to display in the "Domain" drop-down in the login page. [maybe should be implemented in a separate end-point]
My two cents: runPublicQuery shouldn't exist at all. There should be an additional value in org.ovirt.engine.core.common.queries.VdcQueryType.VdcQueryAuthType - Public, which org.ovirt.engine.core.bll.QueriesCommandBase#validatePermissions() should evaluate before even checking if the user is logged in.
*** This bug has been marked as a duplicate of bug 1009843 ***