  Quasi-Noether differential systems are more general than variational systems
and are quite common in mathematical physics. They include practically all
differential systems of interest, at least those that have conservation laws.
In this paper, we discuss quasi-Noether systems that possess
infinite-dimensional (infinite) symmetries involving arbitrary functions of
independent variables. For quasi-Noether systems admitting infinite symmetries
with arbitrary functions of all independent variables, we state and prove an
extension of the Second Noether Theorem. In addition, we prove that infinite
sets of conservation laws involving arbitrary functions of all independent
variables are trivial and that the associated differential system is
under-determined. We discuss infinite symmetries and infinite conservation laws
of two important examples of non-variational quasi-Noether systems: the
incompressible Euler equations and the Navier-Stokes equations in vorticity
formulation, and we show that the infinite sets of conservation laws involving
arbitrary functions of all independent variables are trivial. We also analyze
infinite symmetries involving arbitrary functions of not all independent
variables, prove that the fluxes of conservation laws in these cases are total
divergences on solutions, and demonstrate examples of this situation.
