  In our previous paper, the concept of sub-symmetry of a differential system
was introduced, and its properties and some applications were studied. It was
shown that sub-symmetries are important in decoupling a differential system,
and in the deformation of a system's conservation laws, to a greater extent
than regular symmetries. In this paper, we study the nature of a correspondence
between sub-symmetries and conservation laws of a differential system. We show
that for a large class of non-Lagrangian systems, there is a natural
association between sub-symmetries and local conservation laws based on the
Noether operator identity, and we prove an analogue of the first Noether
Theorem for sub-symmetries. We also demonstrate that infinite conservation laws
containing arbitrary functions of dependent variables can be generated by
infinite sub-symmetries through the Noether operator identity. We discuss the
application of sub-symmetries to the incompressible Euler equations of fluid
dynamics. Despite the fact that infinite symmetries (with arbitrary functions
of dependent variables) are not known for the Euler equations, we show that
these equations possess infinite sub-symmetries. We calculate infinite
sub-symmetries with arbitrary functions of dependent variables for the two-and
three-dimensional Euler equations in the velocity and vorticity formulations
with certain constraints. We demonstrate that these sub-symmetries generate
known series of infinite conservation laws, and obtain new classes of infinite
conservation laws
