We give a systematic study about supersymmetric Euler equations on the smooth dual of the Neveu-Schwarz algebra for N ≤ 3. Let be the inertia operator and , we will show that the N = 2 supersymmetric Euler equation with is local bi-super-Hamiltonian with the freezing point , which is similar to that of the Virasoro algebra , and the N = 3 supersymmetric Euler equation with is local bi-super-Hamiltonian with the freezing point , which is similar to that of the N = 1 Neveu-Schwarz algebra .
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It is well-known that for n = 1, 2, 3, there each exists a unique such extension. For n = 4, there are three kinds of extensions. However, for n > 4, the Lie algebra K0(1|n) has no nontrivial central extension.6,7 We remark that the Virasoro algebra can be viewed as .
In Ref. 35, Popowicz has obtained two super-Hamiltonian operators and . In order to check the compatibility of two Hamiltonian operators, he used computer algebra Reduce and special computer package SUSY2. However, from the above construction, using the central extension and the “frozen-point” method, it is natural to obtain the compatibility.