There are shares, but those shares are not publicly traded over any SEC-regulated exchanges. You still can buy and sell, but you would need to work out the mechanics of the buy/sell with the seller/buyer.
It is essentially as though you incorporated your family business with 1000 (or whatever) shares divided around the family. Over time, different members might transfer those shares privately (e.g., wills, sales, etc.); Superior shares are essentially the same.