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An extended complete Chebyshev system of 3 Abelian integrals related to a non-algebraic Hamiltonian system

In this paper, we study the Chebyshev property of the 3-dimentional vector space E = 〈I0, I1, I2〉, where Ik(h) = ∫ H=h x ky dx and H(x, y) = 1 2 y2+ 1 2 (e−2x+1)−e−x is a non-algebraic Hamiltonian function. Our main result asserts that E is an extended complete Chebyshev space for h ∈ (0, 1 2 ). To this end, we use the criterion and tools developed by Grau et al. in [5] to investigate when a collection of Abelian integrals is Chebyshev.