  The model of the Volterra gyrostat (VG) has not only played an important role
in rigid body dynamics but also served as the foundation of low-order models of
many naturally occurring systems. It is well known that VG possesses two
invariants, or constants of motion, corresponding to kinetic energy and squared
angular momentum, giving oscillatory solutions to its equations of motion. Nine
distinct subclasses of the VG have been identified, two of which the Euler
gyroscope and Lorenz gyrostat are each known to have two constants. This paper
characterizes quadratic invariants of the VG and each of its subclasses,
showing how these enjoy two invariants even when rendered in terms of a
non-invertible transformation of parameters, leading to a transformed Volterra
gyrostat (TVG). If the quadratic coefficients of the TVG sum to zero, as they
do for the VG, the system conserves energy. In all of these cases, the flows
preserve volume. However, physical models where the quadratic coefficients do
not sum to zero are ubiquitous, and characterization of invariants and the
resulting dynamics for this more general class of models with volume
conservation but without energy conservation is lacking. This paper provides
the first such characterization for each of the subclasses of the VG in the
absence of energy conservation, showing how the number of invariants depends on
the number of linear feedback terms. It is shown that the gyrostat with three
linear feedback terms has no invariants. The number of invariants circumscribes
the possible dynamics for these three-dimensional flows, and those without any
invariants are shown to admit rich dynamics including chaos. This gives rise to
a broad class of three-dimensional volume conserving chaotic flows, arising
naturally from model reduction techniques.
