  In the paper, we study the Prandtl system with initial data admitting
non-degenerate critical points. For any index $\sigma\in[3/2, 2],$ we obtain
the local in time well-posedness in the space of Gevrey class $G^\sigma$ in the
tangential variable and Sobolev class in the normal variable so that the
monotonicity condition on the tangential velocity is not needed to overcome the
loss of tangential derivative. This answers the open question raised in the
paper of D. G\'{e}rard-Varet and N. Masmoudi [{\it Ann. Sci. \'{E}c. Norm.
Sup\'{e}r}. (4) 48 (2015), no. 6, 1273-1325], in which the case $\sigma=7/4$ is
solved.
