An exhaustive study is carried out for the determination of the possible types of variation of the second single-ion anisotropy constant K₂. Three types are found and quantified, two of which are anomalous and unknown by now. While they both exhibit a non-monotonic variation with a local extremum between zero and the Curie temperature, one of them is additionally characterized by a zero point and, consequently, by a change of sign of K₂. The value of a unique parameter (the ratio K⁰₃/K^0₂) is vital for the classification of the above types, whereby the value of the first anisotropy constant is completely irrelevant. The dependences for the variation of K₂ are generated with the help of a new parametric method.