  We study the quantum dynamics generated by a two-axis counter-twisting
Hamiltonian from an initial spin coherent state in a spin-$1/2$ ensemble. A
characteristic feature of the two-axis counter-twisting Hamiltonian is the
existence of four neutrally stable and two saddle unstable fixed points. The
presence of the last one is responsible for a high level of squeezing. The
squeezing is accompanied by the appearance of several quantum states of
interest in quantum metrology with Heisenberg-limited sensitivity, and we show
fidelity functions for some of them. We present exact results for the quantum
Fisher information and the squeezing parameter. Although, the overall time
evolution of both changes strongly with the number of particles, we find that
they have regular dynamics for short times. We explain scaling with the system
size by using a Gaussian approach.
