<p>This study investigates the Noether symmetries and geometric properties of the Melvin magnetic spacetime, focusing on the associated first integrals and conservation laws. The analysis begins with a detailed examination of the Noether symmetries, identifying the corresponding infinitesimal generators and their commutation relations. These symmetries yield first integrals that are crucial for understanding the dynamical properties of the spacetime, including Lagrangian invariance, angular momentum, generalized momentum, energy, and linear momentum along the z-axis. The study further delves into the geometric structure of the spacetime, using the vielbein formalism and Cartan structure equations to describe the underlying spin connections. Finally, perturbation and stability analysis is conducted on the effective potential governing the motion of a test particle, comparing unperturbed and perturbed potentials to explore the impact of metric perturbations on the stability of circular orbits. The results provide valuable insights into the stability characteristics of the Melvin magnetic spacetime under small perturbations.</p>