The solutions of static axisymmetric vacuum gravity in dimension 4 are obtained generally by the elegant Weyl method [H. Weyl, Ann. Phys. (4) 54, 117–145 (1917; JFM 46.1303.01)] and in dimension \(d (> 4 )\) static axisymmetric solutions with a rotational \(O(d-2)\) special isometry group can be considered so far only. However, no general analytic solutions were known for dimension greater than four. In this paper, new numerical methods to solve the static axisymmetric vacuum Einstein equations in dimension \(d ( > 4)\) are, first, described. Then it is shown how to apply the technique to black strings in dimension 6 just for various simplifying technical reasons. The compactified non-uniform black string phase connected to the uniform strings at the Gregory-Laflamme critical point is studied as an illustration. In this study, an attempt is made to resolve the thermodynamical properties of the non-uniform string solutions. The solutions with a ratio of maximum to minimum horizon radius up to nine are computed. For a fixed compactification radius, it is noticed that the mass of the non-uniform strings is greater than that of the classically unstable uniform strings. Consequently, they cannot be the end state of the instability. This observation justifies the conjecture of B. Kol [Preprint hep-ph/0207037].