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For a generic $L$-parameter of $U(n)\times U(n)$, it is conjectured thatthere is a unique representation in their associated relevant Vogan $L$-packetwhich produces the unique Fourier-Jacobi model. We investigated this conjecturefor some non-generic $L$-parameters of $U(3)\times U(3)$ and discovered thatthis conjecture is still true for some non-generic $L$-parameter and false forsome non-generic $L$-parameter. In the case when it holds, we specified suchrepresentation under the local Langlands correspondence for unitary group.

On the Fourier-Jacobi model for some endoscopic Arthur packet of $U(3) \times U(3)$ : the non-generic case