The object of the paper is to study the existence of regular solutions of the following integral equation
where and are real parameters. The main result of the paper states that if , then (1) has a positive solution if and only if . This solution can be expressed by the formula . This result answers the question raised by Y. Li [J. Eur. Math. Soc. (JEMS) 6, No. 2, 153–180 (2004; Zbl 1075.45006)]. Some other results concerning (1) are also proved.