2.1. Linear Local Integrals in Planar Rotating Systems

We consider the planar motion of a particle in a stationary gravitational potential in a uniformly rotating frame with angular velocity $\Omega=$ const. A class of potentials of the form

$$U=\frac{1}{2}\left[S(\Psi)+\varphi^{2}-\Omega^{2}\left(x^{2}+y^{2}\right)\right]$$

admits a local integral linear in velocities, expressed as

$$J=A u+B v-\varphi$$

where u and v are the velocity components in the rotating frame, and $\varphi$ is a scalar function of spatial coordinates. A specific example of such a system is constructed, and the domain of admissible motion is explicitly determined. The particle trajectories exhibit coiled structures, and a comparison is made with more realistic galactic potentials (Reference Antonov and Shamshiev2; Reference Shamshiev9).

2.2. Quadratic Local Integrals in Rotating Systems

We investigate local integrals that are quadratic in the velocity components and are valid on invariant surfaces in phase space, even when the potential lacks global symmetries. Such integrals take the general form

$$A u^{2}+B u v+C v^{2}+D u+E v+F=0$$

We identify classes of two-dimensional rotating potentials admitting such integrals. Among them are the following specific forms:

$$U=\frac{\left(2-4 c h^{2} \sigma\right) \xi^{2}}{r^{2}}+\frac{4 L \xi s h \sigma}{r^{3}}-\frac{2 L^{2}-S}{r^{2}}-\Omega^{2} r^{2},\! \mathrm{ and } U=-\frac{\Omega^{2} r^{2}}{2}-\frac{W^{2}}{2 r^{4}}+\frac{S_{0} \sigma}{2 r^{2}}-\frac{\Omega S_{0}}{2 W} r,$$

These potentials define phase-space surfaces on which the velocity field can be directly inferred for specific initial conditions. The complete classification of such cases and corresponding integrals is discussed in (Reference Shamshiev13).

2.3. Three-Dimensional Potentials with Multi-Valued Velocity Fields

In three dimensions, certain stationary potentials permit local integrals that define multivalued velocity fields. In particular, velocity vectors of the form

$$\left.\begin{array}{c}u=\frac{\partial \varPsi}{\partial x} \pm \frac{x}{r} \sqrt{\varphi(r)+2 h}+\Omega y \\[3pt]v=\frac{\partial \varPsi}{\partial y} \pm \frac{y}{r} \sqrt{\varphi(r)+2 h}-\Omega x \\[3pt]w=\frac{\partial \varPsi}{\partial z} \pm \frac{z}{r} \sqrt{\varphi(r)+2 h}\end{array}\right\}$$

lead to the corresponding class of potentials: In the non-rotating case ( $\Omega=0$ ):

$$U(x, y, z)=\frac{1}{2}\left[(\nabla \Phi)^{2}+U_{0}(L)\right],$$

as found in (Reference Antonov and Shamshiev5);

In the rotating frame ( $\Omega=$ const):

$$U=\frac{1}{2}\left[(\nabla \Phi)^{2}+U_{0}(L)+\Omega\left(y \frac{\partial \Phi}{\partial \mathrm{x}}-x \frac{\partial \Phi}{\partial y}\right)\right]$$

as derived in (Reference Shamshiev8). These multi-valued structures generalize conventional box orbits and represent new classes of organized motion in three-dimensional systems.

3.1. Planar Motion in a Rotating Potential

Analytically Integrable Cases. We consider special cases of two-dimensional stationary gravitational fields rotating with constant angular velocity $\Omega$ , for which the existence of a local integral permits full analytical integration of particle trajectories (Reference Antonov and Shamshiev3). Although local integrals generally reduce, but do not eliminate, the complexity of motion, certain specific forms of the potential allow explicit expressions for the trajectories. One such class of potentials is given by:

$$U=\frac{1}{2}\left[S(r)+2 \Omega \chi \exp (-h \sigma)+\chi^{2} r^{-2} \exp (-2 h \sigma)\right]$$

where $S(r), \chi$ , and $\sigma$ are functions or parameters describing the potential structure. In these cases, the equations of motion can be fully integrated, and the orbital paths determined in closed analytical form. While some solutions correspond to trivial configurations, the framework also yields nontrivial, previously unidentified integrable cases, extending the class of analytically tractable models in stellar dynamics.

3.2. Extended Integrability in Three-Dimensional Systems

Building on the notion of local integrals, we explore further conditions under which full integrability can be achieved in three-dimensional rotating systems. A key result, introduced by Antonov and Shamshiev (1992), involves the identification of a local integral that specifies the velocity field for selected initial conditions, thereby reducing the phasespace dimensionality (Reference Antonov and Shamshiev5; Reference Shamshiev11)

We examine cases where this reduction leads to explicit relationships among the spatial coordinates and time along a trajectory, i.e.,

$$x=x(t), \quad y=y(t), \quad z=z(t)$$

via direct integration of the equations of motion. A classification of such potentials is provided, and one representative case is analyzed in detail. These examples highlight the potential of local integrals to reveal hidden symmetries and partially integrable structures even in the absence of global conservation laws (Reference Shamshiev11; Reference Shamshiev13).