I would like an example of a J-structure $(J^A,B)$ which is not acceptable and one that is not 1-sound.

Edit:Let us recall that a structure $J^A_\alpha$ is acceptable if for every limit ordinal $ \xi<\alpha $. $J^A_{\xi+\omega}\models \vert \xi\vert\leq \vert \tau\vert $, whenever $\tau<\xi$ and satisfies ${\mathcal P}(\tau)\cap J^A_{\xi+\omega}\not \subset J^A_\xi $

A structure $J^A_\alpha$ is 1-sound if the 1-standard parameter is a very good parameter.

(I'm using the notation in Zeman's article in the handbook of set theory)

It is known that $J_\alpha$ are acceptable and sound for every ordinal $\alpha$. Moreover, being acceptable and sound is needed for almost all basic results concerning the $J^A_\alpha$ hierarchy.