  Using Hilbert-Burch matrices, we give an explicit description of the
Bia{\l}ynicki-Birula cells on the Hilbert scheme of points on $\mathbb A ^2$
with isolated fixed points. If the fixed point locus is positive dimensional we
obtain an \'etale rational map to the cell. We prove Conjecture 4.2 from
arXiv:2309.06871 which we realize as a special case of our construction. We
also show examples when the construction provides a rational \'etale map to the
Hilbert scheme which is not contained in any Bia{\l}ynicki-Birula cell.
Finally, we give an explicit description of the formal deformations of any
ideal in the Hilbert scheme of points on the plane.
