The number of cyclic configurations of type (v₃) and the isomorphism problem

A configuration of points and lines is cyclic if it has an automorphism which permutes its points in a full cycle. A closed formula is derived for the number of non-isomorphic connected cyclic configurations of type (v_3), i.e., which have v points and lines, and each point/line is incident with exactly 3 lines/points. In addition, a Bays-Lambossy type theorem is proved for cyclic configurations if the number of points is a product of two primes or a prime power.