  We construct various exact analytical solutions of the $SO(3)$ BMN matrix
model that correspond to rotating fuzzy spheres and rotating fuzzy tori.These
are also solutions of Yang Mills theory compactified on a sphere times time and
they are also translationally invariant solutions of the $\mathcal{N} = 1^*$
field theory with a non-trivial charge density. The solutions we construct have
a $\mathbb{Z}_N$ symmetry, where $N$ is the rank of the matrices. After an
appropriate ansatz, we reduce the problem to solving a set of polynomial
equations in $2N$ real variables. These equations have a discrete set of
solutions for each value of the angular momentum. We study the phase structure
of the solutions for various values of $N$. Also the continuum limit where
$N\to \infty$, where the problem reduces to finding periodic solutions of a set
of coupled differential equations. We also study the topology change transition
from the sphere to the torus.
