We give necessary and sufficient conditions for the matrix equations ATX± XTA=B to admit solutions and give their general solution. Because A may be singular, the solution involves the generalized inverse of A. The (-) equation underlies the modern R-matrix approach to completely integrable Hamiltonian systems. This paper provides a new and algorithmic approach to constructing such R-matrices.
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H. W. Braden (1998) studied this question.
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