Abstract
A new approach is adopted to completely classify the Lagrangian associated with the static cylindrically symmetric spacetime metric via Noether symmetries. The determining equations representing Noether symmetries are analyzed using a Maple algorithm that imposes different conditions on metric coefficients under which static cylindrically symmetric spacetimes admit Noether symmetries of different dimensions. These conditions are used to solve the determining equations, giving the explicit form of vector fields representing Noether symmetries. The obtained Noether symmetry generators are used in Noether's theorem to find the expressions for corresponding conservation laws. The singularity of the obtained metrics is discussed by finding their Kretschmann scalar.