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Two different bilinear mappings in PBC

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In pairings-based cryptography, are there any examples of systems where they use two different bilinear mappings. That is, they make use of both $e_1$ and $e_2$ where $e_1$ is a symmetric bilinear mapping from $G_1 \times G_1 \rightarrow G_T$ and $e_2$ is an asymmetric bilinear mapping from $G_1 \times G_2 \rightarrow G_T$. If not, then what are the possible issues in using two bilinear mappings to build say a signature scheme?

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