is known that the strings evolve in a scale-invariant manner [@Albrecht; @Bennett; @Allen]. There are renewed interest of cosmic strings, recently, in relation to the spacetime geometry of the compact extra dimensions of fundamental string theories including branes[@Sarangi:2002yt; @Dvali:2003zj; @Copeland:2003bj]. A detailed investigation[@Jackson:2004zg] suggests that the reconnection probability of this type of strings is considerably suppressed. In this case, the strings in the universe are practically stable. Then, it is a natural question that what is the final state of cosmic strings?

It is well known that the final state of black holes is the Kerr spacetime, which is a stationary state. We guess analogously that the final state of cosmic strings would be a stationary string. The stationary string is defined as the world surface which is tangent to a time-like Killing vector field if one neglects the thickness and the gravitational effects of a string. There are many works on stationary strings in various stationary target spacetimes[@BurdenSRS; @VLS; @FSZH; @Anderson].

We consider infinitely long and stationary rotating strings in the Minkowski spacetime whose dynamics is governed by the Nambu-Goto action in this paper. Though this subject is already studied by Burden and Tassie[@BurdenSRS] in a different motivation, and 