  The Brownian gyrator (BG) is often called a minimal model of a nano-engine
performing a rotational motion, judging solely upon the fact that in
non-equilibrium conditions its torque, angular momentum ${\cal L}$ and angular
velocity $\cal W$ have non-zero mean values. For a time-discretized model,
which is most adapted for the analysis of an essentially discrete-time data
garnered in experiments or numerical simulations, we calculate the previously
unknown probability density functions (PDFs) of ${\cal L}$ and $\cal W$. For
finite time-step $\delta t$, the PDF of ${\cal L}$ has exponential tails and
all moments are therefore well-defined, but the noise-to-signal ratio can
attain big values for small $\delta t$. Conversely, the PDF of ${\cal W}$
exhibits heavy power-law tails and its mean ${\cal W}$ is the only existing
moment. The BG is therefore not an engine in the common sense: it does not
exhibit regular rotations on each run and its fluctuations are not only a minor
nuisance -- on contrary, their effect is completely destructive for the
performance. Our theoretical predictions are confirmed by numerical simulations
and experimental data. We discuss some plausible improvements
