  Motivated by the regularization by noise phenomenon for SDEs we prove
existence and uniqueness of the flow of solutions for the non-Lipschitz
stochastic heat equation $$\frac{\partial u}{\partial
t}=\frac12\frac{\partial^2 u}{\partial z^2} + b(u(t,z)) + \dot{W}(t,z), $$
where $\dot W$ is a space-time white noise on $\mathbb{R}_+\times\mathbb{R}$
and $b$ is a bounded measurable function on $\mathbb{R}$. As a byproduct of our
proof we also establish the so-called path--by--path uniqueness for any initial
condition in a certain class on the same set of probability one. This extends
recent results of Davie (2007) to the context of stochastic partial
differential equations.
