  We introduce a method to determine whether a given generalised quantum
measurement is isolated or it belongs to a family of measurements having the
same prescribed symmetry. The technique proposed reduces to solving a linear
system of equations in some relevant cases. As consequence, we provide a simple
derivation of the maximal family of Symmetric Informationally Complete
measurements (SIC)-POVM in dimension 3. Furthermore, we show that the following
remarkable geometrical structures are isolated, so that free parameters cannot
be introduced: (a) maximal sets of mutually unbiased bases in prime power
dimensions from 4 to 16, (b) SIC-POVM in dimensions from 4 to 16 and (c)
contextuality Kochen-Specker sets in dimension 3, 4 and 6, composed of 13, 18
and 21 vectors, respectively.
