  We consider an overdamped run-and-tumble particle in two dimensions, with
self propulsion in an orientation that stochastically rotates by 90 degrees at
a constant rate, clockwise or counter-clockwise with equal probabilities. In
addition, the particle is confined by an external harmonic potential of
stiffness $\mu$, and possibly diffuses. We find the exact time-dependent
distribution $P\left(x,y,t\right)$ of the particle's position, and in
particular, the steady-state distribution $P_{\text{st}}\left(x,y\right)$ that
is reached in the long-time limit. We also find $P\left(x,y,t\right)$ for a
"free" particle, $\mu=0$. We achieve this by showing that, under a proper
change of coordinates, the problem decomposes into two
statistically-independent one-dimensional problems, whose exact solution has
recently been obtained. We then extend these results in several directions, to
two such run-and-tumble particles with a harmonic interaction, to analogous
systems of dimension three or higher, and by allowing stochastic resetting.
