Angular velocity $\vec{\omega}$ can be defined in terms of velocity $\vec{v}$ and position $\vec{s}$ as:
$$ \vec{\omega} = \frac{\vec{s} \times \vec{v}}{\left\lvert s\right\rvert^2} $$
Constant angular motion therefore obeys the equation:
$$ \frac{\vec{s} \times \vec{v}}{\left\lvert s\right\rvert^2} = \vec{c} $$
What kind of motion obeys a higher order version of the equation?
$$ \vec{\theta} = \int \vec{\omega} \, \mathrm{d} t $$
$$ \vec{c} = \frac{\vec{\theta} \times \vec{\omega}}{\left\lvert \theta \right\rvert^2} $$
I know how to solve differential equations but not vector differential equations.
Constant angular velocity traces out motion like:
$$ \vec{s} = \cos \left(\left\lvert \omega \right\rvert t\right) \hat{a} + \sin \left(\left\lvert \omega \right\rvert t\right) \left(\hat{\omega} \times \hat{a} \right) + b \hat{\omega}$$
A higher order version of this would trace out motion like:
$$ \vec{\theta} = \cos \left(\left\lvert c\right\rvert t\right) \hat{a} + \sin \left(\left\lvert c \right\rvert t\right) \left(\hat{c} \times \hat{a}\right) + b \hat{c}$$
for some constants $\hat{a}$, $\vec{c}$ and b.
Such motion could be visualized as tracing out a circle on a sphere.
Note the third component in the equations. The thing moving in a circular manner can be offset along the axis of rotation and still have a constant angular velocity.