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Yes.

On Hive Engine you stake Leo and Swap.hive

What do I earn from the staking?

There are earnings from the swap fees.
I don't know how much exactly

I don't think you can stake swap.hive on Hive Engine (though you can withdraw it as hive then stake the hive). You can stake Leo.

I don't think there are any direct rewards for just staking Leo, however you can earn Leo curation rewards with your staked Leo. Also, if you delegate hive to leo.voter, you'll get a higher percent return if you don't unstake.

As far as liquidity pools, i don't follow those too closely so I'm not sure what is available.