Liouville Foliations of Confocal Billiard Books with Elliptic Boundaries

For billiard system on each piecewise-planar table with permutations (billiard book) consisting of elliptic disks and annuli, its Fomenko–Zieschang invariant or marked molecule (Reeb graph of the foliation equipped with numerical marks and types of non-degenerate singularities called Fomenko 3-atoms) is calculated. Thus, integrable billiard systems with 2 degrees of freedom from this class are classified in their non-singular energy zones up to Liouville equivalence. An algorithm is presented which is implemented in software and calculates this invariant starting from combinatorial data of the billiard table and its permutations.