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Studies the topology of 2nd-order symmetric tensor fields. Degenerate points are basic constituents of tensor fields. From the set of degenerate points, an experienced researcher can reconstruct a whole tensor field. We address the conditions for the existence of degenerate points and, based on these conditions, we predict the distribution of degenerate points inside the field. Every tensor can be decomposed into a deviator and an isotropic tensor. A deviator determines the properties of a tensor field, while the isotropic part provides a uniform bias. Deviators can be 3D or locally 2D. The triple-degenerate points of a tensor field are associated with the singular points of its deviator and the double-degenerate points of a tensor field have singular local 2D deviators. This provides insights into the similarity of topological structure between 1st-order (or vectors) and 2nd-order tensors. Control functions are in charge of the occurrences of a singularity of a deviator. These singularities can further be linked to important physical properties of the underlying physical phenomena. For a deformation tensor in a stationary flow, the singularities of its deviator actually represent the area of the vortex core in the field; for a stress tensor, the singularities represent the area with no stress; for a Newtonian flow, compressible flow and incompressible flow as well as stress and deformation tensors share similar topological features due to the similarity of their deviators; for a viscous flow, removing the large, isotropic pressure contribution dramatically enhances the anisotropy due to viscosity.