A deflation algorithm was proposed in:
G. Fix and R. Heiberger. An algorithm for the ill-conditioned generalized eigenvalue problem.
SIAM J. Num. Anal., 9:78–88, 1972. 3266.
Sadly, I am not aware of any software that implements the algorithm. Of course, you could use the QZ algorithm, which does not take advantage of symmetry. A routine such as DGGEV returns an eigenvalue as a pair (alpha,beta) such that lambda = alpha/beta. So long as the matrix pair is regular, then if beta is numerically zero, lambda is numerically infinite. For further information, particularly on the singular (non-regular) problem, see:
J. H. Wilkinson. Kronecker’s canonical form and the QZ algorithm. Linear Algebra Appl., 28: