  We discuss static, cylindrically symmetric vacuum solutions of hybrid
metric-Palatini gravity (HMPG), a recently proposed theory that has been shown
to successfully pass the local observational tests and to produce a certain
progress in cosmology. We use HMPG in its well-known scalar-tensor
representation. The latter coincides with general relativity containing, as a
source of gravity, a conformally coupled scalar field $\phi$ and a
self-interaction potential $V(\phi)$. The $\phi$ field can be canonical or
phantom, and accordingly the theory splits into canonical and phantom sectors.
We seek solitonic (stringlike) vacuum solutions of HMPG, that is, completely
regular solutions with Minkowski metric far from the symmetry axis, with a
possible angular deficit. A transition of the theory to the Einstein conformal
frame is used as a tool, and many of the results apply to the general
Bergmann-Wagoner-Nordtvedt class of scalar-tensor theories as well as $f(R)$
theories of gravity. One of these results is a one-to-one correspondence
between stringlike solutions in the Einstein and Jordan frames if the conformal
factor that connects them is everywhere regular. An algorithm for construction
of stringlike solutions in HMPG and scalar-tensor theories is suggested, and
some examples of such solutions are obtained and discussed.
