We construct a planar quartic system and demonstrate that it has at least 26 limit cycles. The vector field is symmetric and integrable, but non-Hamiltonian. The proof is based on a verified computation of zeros of pseudo-Abelian integrals, together with the symmetry properties.
"A Quartic System with Twenty-Six Limit Cycles." Experiment. Math. 20 (3) 323 - 328, 2011.