ELMONT, N.Y. — The Haskell, the Jim Dandy and the Pacific Classic came up more or less in conversation on Sunday at Belmont Park. Monmouth (N.J.), Saratoga (N.Y.) or Del Mar (Calif.) would be the respective tracks. Aug. 1, Aug. 2 and Aug. 22 would be the respective dates. The Travers, on Aug. 29 and also at Saratoga, presents another possibility.

So went the blurry prospectus for the near future of American Pharoah in the hours after he finished dominating the Belmont Stakes and jetting back to Louisville. The first Triple Crown winner in 37 years figures to take his star shine to further races through his 3-year-old season, maybe all the way up to Oct. 31 and the Breeders’ Cup Classic at Keeneland in Lexington, Ky.

“I’m personally taking this responsibility very seriously,” the owner and breeder Ahmed Zayat said. And: “With this horse, we owe it to the sport to continue properly, and as often as we possibly can.” And: “It is my genuine desire, as someone who loves horses, as a fan, to race him as long as I possibly could, at least — at least — until he finishes [as a] 3-year-old.”

Such is the all-around rarity of the horse that trainer Bob Baffert cracked he probably could run him “in three weeks” even if he won’t. Even as the only horse among the entire 3-year-old crop to run all three Triple Crown races, American Pharoah still looked stout to his trainer the next morning, even if slightly, predictably lighter.

“Some of the horses I’ve had in the past, Silver Charm [in 1997], Real Quiet [in 1998], they went through [the Triple Crown] and it was really tough on them,” Baffert said. “I looked at this horse today and for a horse that ran 1 1/2 miles, he looked pretty darned good. He’s just a tough horse and he handled it. We may give him a few days off. He’s an athlete, and we have to keep him moving.”

The crux: “He’ll tell me.”

Said Baffert, “I’ve seen that a lot of horses come out of the Triple Crown and come back, get beat, and I don’t want to see that. I want to make sure that when you see him out there you can feel good about it, that when the gate comes open, [jockey] Victor [Espinoza] will have him in lockdown going around the first turn . . .”

Baffert trainees have won four of the last five Haskell Invitationals, Zayat resides in New Jersey and the Haskell was the first possibility Baffert mentioned, even as he also mentioned his wish to relish the Triple Crown he pursued through three agonizing Belmont Stakes before finally snaring one on Saturday. As Zayat reiterated, he has sold the breeding rights to Coolmore Ashford Stud in Kentucky, but, “I think it’s a huge, huge honor and privilege, and we owe it to the sport to do the right thing.”

At this point, any racing next year at 4 did not seem plausible.

“The problem with a 4-year-old, I think a lot of people, now you’re just running in races, these regular big races, whatever,” Baffert said. “But the [Triple Crown] classics are more, it’s almost like it’s a championship series, and so it’s a best-of-three. And so those other races, you know, they’re big races, it’s more the horse fans, or horse industry, are involved, not really the public. You know, Joe Public, they really, they jump on the classics. They start watching the preps. They make their little future bets, no matter what it is.”

When American Pharoah does race again, he will bring along a seven-race winning streak that began Sept. 3 in the Del Mar Futurity, 25 days after his career began with a fifth-place finish in a $76,000 maiden special weight race on Aug. 9. That outcome led to Baffert equipping American Pharoah with ear plugs. Following a layoff from Sept. 27 to March 14 because of injury, American Pharoah has won five races by a combined 27 3/4 lengths, including the Arkansas Derby by eight, the Preakness by seven, the Rebel Stakes by 6 1/4 and the Belmont Stakes by 5 1/2. As Zayat and Baffert told stories of dazzled farm workers and clockers from early on, Baffert said, “He was a ‘wow’ horse from Day 1.”

As for whether his fresh feat might boost the industry, Baffert said, “It really encourages people to keep breeding horses. It just shows you that you can get lucky and you don’t need a super-fancy pedigree to win these races. And so it gives hope for the little guy. Just like last year, California Chrome, I mean, look at that story, that was a rags-to-riches story, so it can happen, and I think a lot of people get more interested, you know, buying horses, breeding horses and you know, get the numbers back up where they used to be.”