  The main results are the following. We derived the matching conditions for
the spherically symmetric singular hypersurface (in our case it is equivalent
to the world line) in the Weyl$+$Einstein gravity. It was found, that the
residual extrinsic curvature tensor of this surface must be continuous on the
singular hypersurface. The result is the same as that found by Senovilla, it is
dictated by the very possibility to have the double layer, but the jump in the
normal derivative of the radius may be not zero. It was found that in the
presence of the double layer, the matching conditions contain an arbitrary
function, and this result is quite new and very important. One of the
consequences of such freedom is that the trace of the extrinsic curvature
tensor of the singular hypersurface is necessarily equal to zero. We suggested
the physical interpretation for the $S_0^n$ and $S_n^n$ components of the
surface matter energy-momentum tensor of the shell. In General Relativity they
are zero by virtue of the Einstein equations. In the quadratic gravity they are
not necessarily zero. Our interpretation is that these components describe the
energy flow $(S^{0n})$ and the momentum transfer $(S^{nn})$ of the particles
produced by the double layer itself. Moreover, the requirement of the zero
trace of the extrinsic curvature tensor (mentioned above) implies that
$S_n^n=0$, and this fact also support our suggestion, because it means that for
the observer sitting on the shell, the particles will be seen created by pairs,
and the sum of their momentum transfers must zero. We derived also the matching
conditions for the null hypersurface, and this is, again, quite new. We found
that the null-double layer in the Weyl$+$ Einstein gravity does not exist at
all.
