  We consider the Stochastic Differential Equation $X_t = X_0 + \int_0^t
b(s,X_s) ds + B_t$, in $\mathbb{R}^d$. We give an example of a drift $b$ such
that there does not exist a weak solution, but there exists a solution for
almost every realization of the Brownian motion $B$. We also give an explicit
example of a drift such that the SDE has a pathwise unique weak solution, but
path-by-path uniqueness (i.e. uniqueness of solutions to the ODE for almost
every realization of the Brownian motion) is lost. These counterexamples extend
the results obtained in arXiv:2001.02869 to dimension $d=1$.
