  Let ${\cal F}\_\lambda(S^1)$ be the space of tensor densities of degree (or
weight) $\lambda$ on the circle $S^1$. The space ${\cal
D}^k\_{\lambda,\mu}(S^1)$ of $k$-th order linear differential operators from
${\cal F}\_\lambda(S^1)$ to ${\cal F}\_\mu(S^1)$ is a natural module over
$\mathrm{Diff}(S^1)$, the diffeomorphism group of $S^1$. We determine the
algebra of symmetries of the modules ${\cal D}^k\_{\lambda,\mu}(S^1)$, i.e.,
the linear maps on ${\cal D}^k\_{\lambda,\mu}(S^1)$ commuting with the
$\mathrm{Diff}(S^1)$-action. We also solve the same problem in the case of
straight line $\mathbb{R}$ (instead of $S^1$) and compare the results in the
compact and non-compact cases.
