Kapur, who missed his card last year, opened the week with a round of 71 and then added a 68 to be five-under 139.
He was tied 28th, though there were still some players out on the course.
Kapur teed off from the tenth and birdied the 11th, 14th and 15th and then on his second nine, he added a fourth birdie on eighth.
On the first day, he hit five birdies, but also had two bogeys and one double bogey. Peter Lawrie led a group of seven players at the top, all of them being at eight-under 136.
Lawrie carded 70 and 66 and he shared the lead with David Lynn (67-69), Jamie Donaldson (65-71), Pelle Edberg (70-66), Maarten Lafeber (68-68), Simon Wakefield (69-67) and Richard Green.
Shiv Kapur makes cut at Sicily, lying 28th
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