  Given a dynamical system with $m$ independent conserved quantities, we
construct a multi-parameter family of new systems in which these quantities
evolve monotonically and proportionally, and are replaced by $m-1$ conserved
linear combinations of themselves, with any of the original quantities as
limiting cases. The modification of the dynamics employs an exterior product of
gradients of the original quantities, and often evolves the system towards
asymptotic linear dependence of these gradients in a nontrivial state. The
process both generalizes and provides additional structure to existing
techniques for selective dissipation in the literature on fluids and plasmas,
nonequilibrium thermodynamics, and nonlinear controls. It may be iterated or
adapted to obtain any reduction in the degree of integrability. It may enable
discovery of extremal states, limit cycles, or solitons, and the construction
of new integrable systems from superintegrable systems. We briefly illustrate
the approach by its application to the cyclic three-body Toda lattice, driven
from an aperiodic orbit towards a limit cycle.
