Phase trajectories of linear homogeneous autonomous dynamical systems of the third order

The paper correctly states standard facts about 3D linear autonomous systems: the general solution for three distinct real eigenvalues and for one real eigenvalue plus a complex-conjugate pair; and the canonical equations for invariant lines and planes (see the paper’s statements of the general solution and canonical equations ). However, it also makes two substantive errors: (i) it claims that after Gram–Schmidt orthogonalization “these orthogonal vectors define invariant lines – coordinate axes,” which is false in general unless A is normal; Gram–Schmidt does not preserve A-invariance of directions (); and (ii) in the complex-pair case it asserts the presence of “two invariant lines corresponding to complex conjugate roots,” contradicting its own description of a unique invariant plane and the fact that a real 2×2 rotation–dilation block has no real eigenlines (). By contrast, the model’s solution gives the standard, correct real-form reduction for the complex pair, proves uniqueness of the invariant plane inside which there is no invariant line, and uses Gram–Schmidt only to define a plotting frame without asserting invariance. Therefore, the model is correct while the paper contains material errors.