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=hx^kydx and H(x,y)=12y^2+12(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∈(0,12). 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.