  A linearly constrained framework in $\mathbb{R}^d$ is a point configuration
together with a system of constraints which fixes the distances between some
pairs of points and additionally restricts some of the points to lie in given
affine subspaces. It is globally rigid if the configuration is uniquely defined
by the constraint system, and is rigid if it is uniquely defined within some
small open neighbourhood. Streinu and Theran characterised generic rigidity of
linearly constrained frameworks in $\mathbb{R}^2$ in 2010. We obtain an
analagous characterisation for generic global rigidity in $\mathbb{R}^2$. More
precisely we show that a generic linearly constrained framework in
$\mathbb{R}^2$ is globally rigid if and only if it is redundantly rigid and
`balanced'. For generic frameworks which are not balanced, we determine the
precise number of solutions to the constraint system whenever the underlying
rigidity matroid of the given framework is connected. We also obtain a stress
matrix sufficient condition and a Hendrickson type necessary condition for a
generic linearly constrained framework to be globally rigid in $\mathbb{R}^d$.
