This work describes the development of a model using a zigzag chain of weakly-coupled ball resonators with Neumann boundary conditions. The chain is assumed to be constructed of identical resonators connected through point-like apertures. The connecting points are described by their delta-coupling with a constant intensity. The model is based on the theory of self-adjoint extensions of symmetrical operators. Due to effectively one-dimensional joints, the 3D problem can be solved with assistance from the transfer matrix approach. This allows us to study the spectrum of the physical system. In particular, it is proven that the discrete spectrum of direct zigzag chain is empty while bending deformation leads to the appearance of non-empty discrete spectrum. In addition, the continuous spectrum has band structure. With the help of asymptotic study, we obtain the dependence of the spectrum structure on the geometrical and physical parameters of the system: zigzag angle, bend angle and coupling intensity.