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If it is a dunlop style 662 x38c should work or called 700c which is a modern 28 inch. Lots of style tires out there. Try and get them in all black or off white without a lot of markings on them if you can.
I agree with all of the above answers. My experience is that there are inconsistencies as far as the actual ISO of the rim. On some a 27 1/4 ISO 630 will fit, others won't. I've gotten 700's ISO 622 on others w/ a lot of force, found that the tires of the better manufacturers have a little more wiggle room. If they ever come out w/ an ISO 626, I think it may fit perfectly.