  We consider weighted ray-transforms $P\_W$ (weighted Radon transforms along
straight lines) in $\mathbb{R}^d, \, d\geq 2,$ with strictly positive weights
$W$. We construct an example of such a transform with non-trivial kernel in the
space of infinitely smooth compactly supported functions on $\mathbb{R}^d$. In
addition, the constructed weight $W$ is rotation-invariant continuous and is
infinitely smooth almost everywhere on $\mathbb{R}^d \times \mathbb{S}^{d-1}$.
In particular, by this construction we give counterexamples to some well-known
injectivity results for weighted ray transforms for the case when the
regularity of $W$ is slightly relaxed. We also give examples of continous
strictly positive $W$ such that $\dim \ker P\_W \geq n$ in the space of
infinitely smooth compactly supported functions on $\mathbb{R}^d$ for arbitrary
$n\in \mathbb{N}\cup \{\infty\}$, where $W$ are infinitely smooth for $d=2$ and
infinitely smooth almost everywhere for $d\geq 3$.
