  A Lie system is a system of differential equations admitting a superposition
rule, i.e., a function describing its general solution in terms of any generic
set of particular solutions and some constants. Following ideas going back to
the Dirac's description of constrained systems, we introduce and analyse a
particular class of Lie systems on Dirac manifolds, called Dirac--Lie systems,
which are associated with `Dirac--Lie Hamiltonians'. Our results enable us to
investigate constants of the motion, superposition rules, and other general
properties of such systems in a more effective way. Several concepts of the
theory of Lie systems are adapted to this `Dirac setting' and new applications
of Dirac geometry in differential equations are presented. As an application,
we analyze traveling wave solutions of Schwarzian equations, but our methods
can be applied also to other classes of differential equations important for
Physics.
