The move is seen as part of Ukraine's efforts to reinforce its army as soldiers struggle to hold positions against Russia
Cook, 28, was a Monday qualifier into the US PGA Tour event in 2015, when he powered into contention in his second tour start before three late bogeys on Sunday saw him fade.
He recalled the week, which included a third round played alongside Phil Mickelson, as "a great experience" but added: "I think I kind of want to redeem myself."
Cook, who earned his tour card in 2017 and won the RSM Classic as a rookie, got that bid off to a rousing start on the par-72 Golf Club of Houston course.
He hit 12 of 14 fairways and hit all 18 greens in regulation, rolling in a 29-foot birdie putt at the seventh before sticking his second shot at the par-five eighth 12 feet from the pin and draining the eagle putt.
Those were the highlights as he played five holes from the fifth through the ninth in six-under, a sparkling run that was sandwiched between birdies at the third and 13th.
He said a lesson via video with his swing coach had helped him iron out a few things.
"The last few weeks it's kind of been a little iffy with the ball-striking," he said. "Kind of got in a groove right now and just going to work on that and try to dial it in a little bit more."
Talor Gooch's 10 birdies included a 63-footer at the par-three ninth. He had two bogeys but his three birdies in his last four holes included back-to-back birdies to finish.
He and Cook were one stroke in front of Austrian Sepp Straka, who teed off on 10 and birdied his first four holes.
Straka finished with nine birdies and two bogeys.
It was a further stroke back to 2017 Houston Open winner Russell Henley, Lanto Griffin and Tyler McCumber on 66.
It was a good day for Houston native Cole Hammer, the second-ranked amateur in the world who shook off an early double-bogey to card a 67, joining a group of seven players that also included China's Zhang Xinjun and Ireland's Seamus Power.
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