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The aim of this paper is to study Sasakian immersions of (non-compact)complete regular Sasakian manifolds into the Heisenberg group and into $\mathbb{B}^N\times \mathbb{R}$ equipped with their standard Sasakianstructures. We obtain a complete classification of such manifolds in the$\eta$-Einstein case.

$\eta$-Einstein Sasakian immersions in non-compact Sasakian space forms