  We consider Kasner-type static, cylindrically symmetric interior string
solutions in the $f\left(R,L_m\right)$ theory of modified gravity. The physical
properties of the string are described by an anisotropic energy-momentum tensor
satisfying the condition $T_t^t=T_z^z$; that is, the energy density of the
string along the $z$-axis is equal to minus the string tension. As a first step
in our study we obtain the gravitational field equations in the
$f\left(R,L_m\right)$ theory for a general static, cylindrically symmetric
metric, and then for a Kasner-type metric, in which the metric tensor
components have a power law dependence on the radial coordinate $r$. String
solutions in two particular modified gravity models are investigated in detail.
The first is the so-called "exponential" modified gravity, in which the
gravitational action is proportional to the exponential of the sum of the Ricci
scalar and matter Lagrangian, and the second is the "self-consistent model",
obtained by explicitly determining the gravitational action from the field
equations under the assumption of a power law dependent matter Lagrangian. In
each case, the thermodynamic parameters of the string, as well as the precise
form of the matter Lagrangian, are explicitly obtained.
