In this paper, we find all nonstatic plane symmetric spacetime metrics whose corresponding Lagrangians possess Noether symmetries. The set of determining equations is analyzed through a Maple algorithm that restricts the metric coefficients to satisfy certain conditions. These restrictions on metric coefficients, while using them to solve the determining equations, give rise to a number of plane symmetric metrics admitting 4-, 5-, 6-, 7-, 8-, 9-, 11-, and 17-dimensional Noether algebras. The Noether theorem is used to find a conserved quantity corresponding to each Noether symmetry. Some physical implications are discussed by finding bounds for different energy conditions for the obtained metrics.

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.

In this paper, we have explored Noether symmetries for the Lagrangian corresponding to the Lemaitre-Tolman-Bondi (LTB) spacetime metric via a Rif tree approach. Instead of the frequently used method of directly integrating the Noether symmetry equations, a MAPLE algorithm is used to convert these equations to the reduced involutive form (Rif). The interesting feature of this algorithm is that it provides all possible metrics admitting different dimensional Noether symmetries. These metrics are given in the form of branches of a tree, known as a Rif tree. These metrics are used to solve the determining equations and the explicit form of symmetry vector fields are found, giving 4, 5, 6, 7, 8, 9, 11, and 17-dimensional Noether algebras. To add some physical implications, Einstein’s field equations are used to find the stress-energy tensor for all the explicitly known metrics, and the parameters appearing in the metrics are used to find bounds for different energy conditions.