  We equip the regular Fr\'echet Lie group of invertible, odd-class, classical
pseudodifferential operators $Cl^{0,*}_{odd}(M,E)$ -- in which $M$ is a compact
smooth manifold and $E$ a (complex) vector bundle over $M$ -- with
pseudo-Riemannian metrics, and we use these metrics to introduce a large class
of rigid body equations. We adapt to our infinite-dimensional setting Manakov's
classical observation on the integrability of Euler's equations for the rigid
body, and we show that our equations can be written in Lax form (with
parameter) and that they admit an infinite number of integrals of motion. We
also prove the existence of metric connections, we show that our rigid body
equations determine geodesics on $Cl^{0,*}_{odd}(M,E)$, and we present rigorous
formulas for the corresponding curvature and sectional curvature. Our main tool
is the theory of renormalized traces of pseudodifferential operators on compact
smooth manifolds without boundary.
