  In the paper, we study the three-dimensional Prandtl equations without any
monotonicity condition on the velocity field. We prove that when one tangential
component of the velocity field has a single curve of non-degenerate critical
points with respect to the normal variable, the system is locally well-posed in
the Gevrey function space with Gevrey index in $]1, 2].$ The proof is based on
some new observation of cancellation mechanism in the three space dimensional
system in addition to those in the two-dimensional setting obtained in
[1,7,19,22].
