  SgrA$^*$, the supermassive black hole candidate at the Galactic Center,
exhibits flares in the X-ray, NIR, and sub-mm bands that may be interpreted
within a hot spot model. Light curves and images of hot spots orbiting a black
hole are affected by a number of special and general relativistic effects, and
they can be potentially used to check whether the object is a Kerr black hole
of general relativity. However, in a previous study we have shown that the
relativistic features are usually subdominant with respect to the background
noise and the model-dependent properties of the hot spot, and eventually it is
at most possible to estimate the frequency of the innermost stable circular
orbit. In this case, tests of the Kerr metric are only possible in combination
with other measurements. In the present work, we consider a class of non-Kerr
spacetimes in which the hot spot orbit may be outside the equatorial plane.
These metrics are difficult to constrain from the study of accretion disks and
indeed current X-ray observations of stellar-mass and supermassive black hole
candidates cannot put interesting bounds. Here we show that near future
observations of SgrA$^*$ may do it. If the hot spot is sufficiently close to
the massive object, the image affected by Doppler blueshift is brighter than
the other one and this provides a specific observational signature in the hot
spot's centroid track. We conclude that accurate astrometric observations of
SgrA$^*$ with an instrument like GRAVITY should be able to test this class of
metrics, except in the more unlikely case of a small viewing angle.
