  The explicit formulation of the general inverse problem on conservation laws
is presented for the first time. In this problem one aims to derive the general
form of systems of differential equations that admit a prescribed set of
conservation laws. The particular cases of the inverse problem on first
integrals of ordinary differential equations and on conservation laws for
evolution equations are studied. We also solve the inverse problem on
conservation laws for differential equations admitting an infinite dimensional
space of zeroth-order conservation-law characteristics. This particular case is
further studied in the context of conservative first-order parameterization
schemes for the two-dimensional incompressible Euler equations. We exhaustively
classify conservative first-order parameterization schemes for the
eddy-vorticity flux that lead to a class of closed, averaged Euler equations
possessing generalized circulation, generalized momentum and energy
conservation.
