During a combinatorial search for more well-integrable geometries (ha ha), the following differential equation turned up: 8*(2 - 3*x)*T - d/dx( x*(1 - x)*(25 - 27*x)*T' ) = 0. As the period constraint of a Hamiltonian: 2H = r^2 - (8/25)*r^4 + (18/625)* r^6 - (2/625)* r^6 *Cos[6*z] . ( in (r,z) polar coordinates ) The integer coefficients of the period expansions are: T@0=1, 16, 330, 7360, 170650, 4050816, 97699056, 2383814400,... T@(25/27)=1, 84, 16830, 3971640, 1030948650, 282553385184, 80264956143456,... T@1=1, 4, 30, 280, 2890, 31584, 358176, 4168560, 49455450, 595480600,... and the integer coefficients of nome expansions are: q@0=0, 1, 40, 1746, 79632, 3730134, 177917808, 8598649428, 419770317216,... q@(25/27)=0, 1, 285, 122346, 59567918, 31213994709, 17129120744817,... q@(1)=0, 1, 65, 5550, 540250, 56586875, 6208576875, 703359012500, ... Only T@1 is in OEIS: https://oeis.org/A274665, and it comes from a list by Bostan et al. see ref. [1,2] entry SS[16]. The authors are interested in connections to physics, so I think it's right in line to mention this alternative interpretation from Hamiltonian mechanics. Entries SS[2] and SS[15] have similar interpretations in terms of Hamiltonian functions with elliptic level curves. I checked the list given on J.A. Weil's website, and didn't find any other matches with my own records. It would be nice to have a more complete list of similar geometries, but so far it has not been too easy to make progress (stronger, faster algorithms are always needed). --Brad [1] https://arxiv.org/pdf/1507.03227.pdf [2] http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/3var/3var_ORD...