Let us consider a classical mechanical system of N particles in a constant external field. We have 3N coordinates and 3N velocities, so totally 6N unknown variables. We have 6N ordinary differential equations of the first order for them or 3N equations of the second order. It is known that such a system has 6N-1 conserved quantities, see Landau-Lifshits, V. 1, Classical Mechanics, Chapter about Conservation Laws. In absence of symmetries these conserved quantities are rather complicated analytical expressions.
In case of some symmetries one can construct conserved quantities additive on particles. Everybody knows about total energy, momentum, and angular momentum conservation laws. But the number of conserved quantities remains to be 6N-1. It means the symmetries may help us construct "less messy" integrals of motion than in general case. We obtain simplification of analytical expressions but not new integrals of motion in case of symmetries.
The question I wrote in title has, in my opinion, the following answer: Symmetries do not lead to additional conservation laws but to simplification of existent ones. To a great extent it is due to simplification of equation system in case of symmetries.
An example of integrals of motion in 1D case: