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Versal Deformations of Infinitesimally Symplectic Transformations with Antisymplectic Involutions Yieh-Hei Wan ABSTRACT Normal forms for versal unfoldings of linear Hamiltonian systems anti-commute with an anti-symplectic involution are given in this paper. They can be derived from suitable chosen versal unfoldings of linear Hamiltonians without an involution. The results are expressed in an alternative basis and in a symplectic basis compatible with this involution. Descriptions of unfoldings of codimension < 2 are given for an illustration. §1. Introduction Consider the phase portraits of a Hamiltonian system ~ = L(/~)x + O( Ixl 2) near zero, which depends on some parameters #. Often there exists a fixed involution p (p2 = 1) which reverses this Hamiltonian system (cf. Mackay ). The involution p can be regarded as a symmetry on this system involving space and time. Recognizing this symmetry, may allow us to analyze this system more easily. Indeed, the number of parameters can be reduced. As a first step in carrying out such an investigation, one needs to find (a) a normal form for a linear Hamiltonian system (with or without a fixed involution). (b) a normal form for a typical family of linear Hamiltonian systems. *RESEARCH SUPPORTED BY NATIONAL SCIENCE FOUNDATION UNDER GRANT DMS--8901645