  Complex techniques of general relativity are used to determine \emph{all} the
states in the two and three dimensional momentum spaces in which the equality
holds in the uncertainty relations for the non-commuting basic observables of
Euclidean invariant elementary quantum mechanical systems, even with non-zero
intrinsic spin. It is shown that while there is a 1-parameter family of such
states for any two components of the angular momentum vector operator with any
angle between them, such states exist for a component of the linear and angular
momenta \emph{only if} these components are orthogonal to each other and hence
the problem is reduced to the two-dimensional Euclidean invariant case. We also
show that the analogous states exist for a component of the linear momentum and
of the centre-of-mass vector \emph{only if} the angle between them is zero or
an acute angle. \emph{No} such state (represented by a square integrable and
differentiable wave function) can exist for \emph{any} pair of components of
the centre-of-mass vector operator. Therefore, the existence of such states
depends not only on the Lie algebra, but on the choice for its generators as
well.
