Classes, orders, families, genera, and species which may not be imported, transported, possessed, or released alive in this state solely because of concern for the welfare of the animal may be possessed under permit when the owner can demonstrate that such animal was legally acquired and possessed in California before the effective date of this section. The department may require the owner of an animal which may be possessed under this section to mark or otherwise identify such animal and progeny, so as not to endanger the welfare of that animal, to the satisfaction of the department. The owner shall not transfer such animal or progeny to any other person without prior approval of the department.