We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian planar polynomial vector fields of degree 3 symmetric with respect to the $ x- $axis having a nilpotent saddle at the origin.
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Figure 4. The black lines correspond to the curves where the number of finite singular points can change (see Figure 3 for more details), and the gray lines correspond to the curves where the connection of saddles can occur. The connection with the saddle at the origin can occur on the upper gray line, and the connection between two saddles different from the origin can occur on the lower gray line
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