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(a) Under this subchapter, the ordinance of the governing body of a city or town may contain therein a classification of vehicles by types and the rate of tax levy, stated in dollars and cents, to be collected from the owners of the vehicles coming within these classifications.
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(b) No such classification shall, at the time of the adoption of the ordinance, include any vehicle for the use of which a state tax or fee for the registration or licensing of motor vehicles is not, at the time, levied upon the owner.
