Conservation laws of evolution systems with constraints

Evolution systems of equations in partial derivatives in the presence of differential constraints in the space variables are considered. It is known that when differential constraints are absent evolution systems may posses only conservation laws, written as the continuity equation valid on solutions of the system. It is shown that when differential constraints are present conservation laws of the different type are possible. Namely, when the previous conservation laws are written as differential forms, the degree of these forms is equal to the number of the space variables, while the new conservation laws are written as differential forms of the degree that is less than the number of the space variables by one. Moreover, if the constraints are written as the continuity equations than the new conservation laws are always present, and their number is equal to the number of constraints. As a characteristic example the Maxwell equations are considered.