Problem conditions to use Laplacian solvers
It's a good question. You can use a Laplacian solver if $A$ is symmetric and diagonally semi-dominant (SDD). This is the subject of Theorem 9.2 in your reference book from Vishnoi. A good exposition of the proof (which was originally by Gremban) can be found in Appendix A of the paper by Kelner, Orecchia, Sidford and Zhu (arXiv:1301.6628).
A recent result by Kyng and Zhang (arXiv:1705.02944) suggests that current Laplacian solvers are probably restricted to this class.