Circular restricted three-body problem

Problem

Let $\mu \in (0, \frac{1}{2}]$. Two bodies, the primaries, have masses $1 - \mu$ and $\mu$. They move on circles around their common centre of mass, at distance $1$ from each other and with angular velocity $1$. This is a circular solution of the two-body problem, in units where the gravitational constant times the total mass is $1$. A third body of negligible mass moves in their gravitational field and does not affect them. Use the rotating frame in which the primaries stay at $(-\mu, 0, 0)$ and $(1 - \mu, 0, 0)$. In this frame the position $(x, y, z)$ of the third body satisfies $$x'' - 2 y' = \frac{\partial \Omega}{\partial x}, \qquad y'' + 2 x' = \frac{\partial \Omega}{\partial y}, \qquad z'' = \frac{\partial \Omega}{\partial z},$$ where $$\Omega(x, y, z) = \frac{x^2 + y^2}{2} + \frac{1 - \mu}{r_1} + \frac{\mu}{r_2},$$ and $r_1 = \sqrt{(x + \mu)^2 + y^2 + z^2}$ and $r_2 = \sqrt{(x - 1 + \mu)^2 + y^2 + z^2}$ are the distances to the primaries. Given the initial position and velocity of the third body, describe its motion for all time.