New 2D integrable families with a quartic second invariant

The method introduced by the present author in 1986 still proves most effective in constructing integrable 2-D Lagrangian systems, which admit in addition to the energy another integral of motion that is polynomial in velocities. In a previous article (J. Phys. A: Math. Gen., 39, 5807-5824, 2006) we constructed a system, which admits a quartic complementary integral. This system, called by us "master", is the largest known, as it involves 21 parameters, and contains, as special cases of it, almost all previously known systems of the same type that admit a quartic integral. In the present note we generalize the method we used before to construct new several-parameter systems that are not special cases of the master system. A new system involving 16 parameters is introduced and a special case of it admits interpretation in a problem of rigid body dynamics. It gives a unification of certain special versions of known classical integrable cases due to Kovalevskaya, Chaplygin and Goryachev and other cases recently introduced by the present author.