The notable exceptions are The Bachelor and the Bobby-Soxer (which is now owned by Warner Bros., via-Turner Entertainment) and Gone to Earth.
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As with all PD MGM feature-length films produced by the studio itself (and possibly a few they merely distributed), the original film elements are now owned by Turner Entertainment, with distribution rights handled by Warner Bros. (who spoofed the title in one of their 1954 short subject cartoons, Claws for Alarm).
Warner Bros. also currently owns other titles with Presley via purchasing Turner Entertainment, including Presley's MGM films and National General Pictures' Charro!.
For example, Strapless is distributed by Anchor Bay Entertainment, while Warner Home Video (distributing for Turner Entertainment) owns the DVD rights to A Touch of Class.