Symplectic circle actions with isolated fixed points

Consider a symplectic circle action on a closed symplectic manifold $M$ with non-empty isolated fixed points. Associated to each fixed point, there are well-defined non-zero integers, called \emph{weights}. We prove that the action is Hamiltonian if the sum of an odd number of weights is never equal to the sum of an even number of weights (the weights may be taken at different fixed points). Moreover, we show that if $\dim M=6$, or if $\dim M=2n \leq 10$ and each fixed point has weights $\{\pm a_1, \cdots, \pm a_n\}$ for some positive integers $a_i$, the action is Hamiltonian if the sum of three weights is never equal to zero. As applications, we recover the results for semi-free actions, and for certain circle actions on six-dimensional manifolds. Finally, we prove that if there are exactly three fixed points, $M$ is equivariantly symplectomorphic to $\mathbb{CP}^{2}$