We consider a real linear Hamiltonian system with constant coefficients, which depend on several real parameters. The necessary and sufficient conditions for the stability of stationary solutions of this system for fixed values of the parameters are formulated and proved. A method for computing a set of all the values, for which this solution is stable, is proposed. Application of the method is demonstrated using a gyroscopic problem, described by the Hamiltonian system with four degrees of freedom and with three parameters. Computer algebra (Groebner basis) and Power Geometry are used in the calculations. The four- parameter generalization of this problem does not contain principally new difficulties
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