  We study the problem of minimising the connection time between
non-equilibrium steady states of the Brownian Gyrator. This is a paradigmatic
model in non-equilibrium statistical mechanics, an overdamped Brownian particle
trapped in a two-dimensional elliptical potential, with the two degrees of
freedom $(x,y)$ coupled to two, in principle different, thermal baths with
temperatures $T_x$ and $T_y$, respectively. Application of Pontryagin's Maximum
Principle reveals that shortest protocols belong to the boundaries of the
control set defined by the limiting values of the parameters $(k,u)$
characterising the elliptical potential. We identify two classes of optimal
minimum time protocols, i.e. brachistochrones: (i) regular bang-bang protocols,
for which $(k,u)$ alternatively take their minimum and maximum values allowed,
and (ii) infinitely degenerate singular protocols. We thoroughly investigate
the minimum connection time over the brachistochrones in the limit of having
infinite capacity for compression. A plethora of striking phenomena emerge:
sets of states attained at null connection times, discontinuities in the
connection time along adjacent target states, and the fact that, starting from
a state in which the oscillators are coupled, uncoupled states are impossible
to reach in a finite time.
