ArealLiouvilledomainisaLiouvilledomainwithanexactanti-symplecticinvolution.TheauthorscallarealLiouvilledomainuniruledifthereexistsaninvariantfiniteenergyplanethrougheveryrealpoint.Asymptotically,aninvariantfiniteenergyplaneconvergestoasymmetricperiodicorbit.Inthisnote,theyworkoutacriterionwhichguaranteesunirulednessforrealLiouvilledomains.