A FAMILY OF COMPACT SEMITORIC SYSTEMS WITH TWO FOCUS-FOCUS SINGULARITIES

摘要

About 6 years ago, semitoric systems were classified by Pelayo & V~u Ngo. c by means of five invariants. Standard examples are the coupled spin oscillator on S~2 × R~2 and coupled angular momenta on S~2 × S~2, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a 6-parameter family of integrable systems on S~2 × S~2 and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.