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C. In Case 3: 1. four two-dimensional orbits homeomorphic to a plane, for C\C2 ^ 0; 2. 4) and a closed curve, for C\C2 = 0, C\ + 7^ 0. In this case every component consists of one one-dimensional nonclosed orbit and four two-dimensional orbits (planes), adjacent to it; 3. a direct product of two “crosses”, for C\ = C2 = 0. In this case L(C \,C 2) consists of a singular point, 8 one-dimensional nonclosed orbits adjacent to it, and 16 two-dimensional orbits (planes), such that each of them contains a singular point and two one-dimensional orbits in its closure.

These planes are obtained as products of a singular point of the plane L\ and the foliation of L2, and vice versa, with interchanging L\ and L2. One-dimensional orbit foliations on these planes coincide with the foliations into trajectories of Hamiltonian vector fields on the corresponding planes Li, L2. In the case of focus-focus we have E = 0. Now we can give a linear classification of simple LPAs. 1. Two simple LPAs Sb and if their singular points are of the same type. are linearly equivalent if and only P roof.

X n at the point p is n-dimensional and contains an operator with simple eigenvalues. It is easy to see that L\ may play a role of this operator for a simple singular point of the IHVF. A special representation of a vector field in a neighborhood of a singular point, called its normal form [42], plays an important role in the local study of the singular point. For Hamiltonian vector fields, when symplectic coordinate transformations are used, it is more convenient to bring into normal form not the vector field but the Hamilton function itself.