  This paper is dedicated to approximate controllability for Grushin equation
on the rectangle $(x,y) \in (-1,1) \times (0,1)$ with an inverse square
potential. This model corresponds to the heat equation for the Laplace-Beltrami
operator associated to the Grushin metric on $\mathbb{R}^2$, studied by Boscain
and Laurent. The operator is both degenerate and singular on the line $\{ x=0
\}$.
  The approximate controllability is studied through unique continuation of the
adjoint system. For the range of singularity under study, approximate
controllability is proved to hold whatever the degeneracy is.
  Due to the internal inverse square singularity, a key point in this work is
the study of well-posedness. An extension of the singular operator is designed
imposing suitable transmission conditions through the singularity.
  Then, unique continuation relies on the Fourier decomposition of the 2D
solution in one variable and Carleman estimates for the 1D heat equation solved
by the Fourier components. The Carleman estimate uses a suitable Hardy
inequality.
