Ho$\check{ \textbf{ r}}$ava proposed a non-relativistic renormalizable theory of gravitation, which is reduced to general relativity (GR) in large distances (infra-red regime (IR)). It is believed that this theory is an ultra-violet (UV) completion for the classical theory of gravitation. In this paper, after a brief review of some fundamental features of this theory, we investigate it for a static cylindrical symmetric solution which describes \emph{ Cosmic string} as a special case. We have also investigated some possible solutions, and have seen that how the classical GR field equations are modified for generic potential $V (g)$. In one case there is an algebraic constraint on the values of three coupling constants. Finally as a pioneering work we deduce the most general \emph{ cosmic string} in this theory. We explicitly show that how the \emph{ coupling constants} distort the mass parameter of \emph{ cosmic string}. We deduce an explicit function for mass per unit length of the space-time as a function of the \emph{ coupling constants}. We compare this function with another which Aryal et al [58] have found in GR.Also we calculate the self-force on a massive particle near Ho$\check{ \textbf{ r}}$ava-Lifshitz straight string and we give a typical order for the \emph{ coupling constants} $g_{ 9}$. This order of magnitude proposes a cosmological test for validity of this theory.
Categories: gr-qc hep-th math-ph math.CA math.MP