"Inverse-GITA" for an inverse problem of the Birkhoff-Gustavson normal-form method

When a dynamical system is studied successfully by the Birkhoff-Gustavson n ormal-form (BGNF) method, an "inverse problem" can be formulated as follows : How can we specify explicitly a class of dynamical systems that reduce to the same BGNF up to a certain order? To solve this inverse problem, a symb olic computing procedure called "the inverse-GITA" is proposed. It is reali zed by inverting GITA, the symbolic computing procedure to calculate BGNF. As an application of the inverse-GITA, it is shown that the inverse-GITA pr ovides a class of integrable Hamiltonian systems of the Liouville type that share the same BGNF Hamiltonian with the regularized system of a planar hy drogen atom displaying a linear Stark effect.