  In the letter we give new symmetries for the isospectral and non-isospectral
Ablowitz-Ladik hierarchies by means of the zero curvature representations of
evolution equations related to the Ablowitz-Ladik spectral problem. Lie
algebras constructed by symmetries are further obtained. We also discuss the
relations between the recursion operator and isospectral and non-isospectral
flows. Our method can be generalized to other systems to construct symmetries
for non-isospectral equations.
