  In the first part of the paper we continue the study of solutions to
Schr\"odinger equations with a time singularity in the dispersive relation and
in the periodic setting. In the second we show that if the Schr\"odinger
operator involves a Laplace operator with variable coefficients with a
particular dependence on the space variables, then one can prove Strichartz
estimates at the same regularity as that needed for constant coefficients. Our
work presents a two dimensional analysis, but we expect that with the obvious
adjustments similar results are available in higher dimensions.
