Generalization of a Max Noether's Theorem

Max Noether’s Theorem asserts that if $\text{ww}$ is the dualizing sheaf of a nonsingular nonhyperelliptic projective curve then the natural morphisms ${\text{Sym}}^n{H}^0(\omega )\to H^0({\omega }^n)$ are surjective for all $n\ge 1$. This is true for Gorenstein nonhyperelliptic curves as well. We prove this remains true for nearly Gorenstein curves and for all integral nonhyperelliptic curves whose non-Gorenstein points are unibranch. The results are independent and have different proofs. The first one is extrinsic, the second intrinsic.