In a nutating engine (or more generally a rotary displacement device) the cycling of combustion occurs between inner and outer spherical surfaces. The combustion chambers are additionally defined only by the surfaces of teeth of specially designed gears. These gears are the Rotator (2), two consecutive of some number of free-planetary gears (3A-E), and the lobed Nutating Member (1). The latter is enjoined to execute precessional rotation relative to the former and maybe both affixed with counterweights (11A,B; 12A,B) and subjected to reverse-English transforming the precessional rotation to a stress-free mode in both the Newtonian and Eulerian sense. The insertion of optional butterfly-shaped plugs (4A-E, 5A-E, 8A-E) boosts the compression ratio. Truly, no past engine possesses the characteristics of the instant invention.