New family of Abelian integrals satisfying Chebyshev property

Abstract This paper is motivated by the Chebyshev property of a family of Abelian integrals along non-algebraic level ovals. A new criterion is established to study the Chebyshev property of these Abelian integrals. As its applications, we give three examples to show that the ordered sets defined by Abelian integrals are ECT systems. Moreover, the number of the Abelian integrals can be an arbitrary positive integer.