The primitive solutions of certain generalized Fermat equations, i.e., Diophantine equations of the form $Ax^p+By^q = Cz^r$, can be studied as integral points on certain stacky curves. In a recent paper by Bhargava and Poonen, an explicit example of such a curve of genus $\frac{1}{2}$ violating local-global principle for integral points was given. However, a general description of stacky curves failing the local-global principle is unknown. In this talk, I will discuss our work on finding the primitive solutions to equation of the form when $(p, q, r) = (2,2,n)$ by studying local-global principles for integral points on stacky curves constructed from such equations. The talk is based on a joint work with Juanita Duque-Rosero, Christopher Keyes, Andrew Kobin, Manami Roy and Soumya Sankar.