Why does Lyra use Synthetix?
This page explains why Lyra integrates with Synthetix
There are two main reasons Lyra uses Synthetix, managing collateral and hedging LPs.

Managing Collateral

When you trade options in Lyra, you are trading against an AMM which has a pool of collateral. When options are purchased, the AMM needs to lock enough collateral to payout the contract if it expires in-the-money. Calls are collateralised with the base asset (e.g. sETH) and puts are collateralised with the quote asset (sUSD for all markets).
sUSD is the asset stored in each liquidity pool. When puts are purchased, the AMM simply needs to lock sUSD. However, when calls are purchased, the AMM needs to purchase the base asset to ensure it can cover the payout. Synthetix provides a zero slippage exchange which means the AMM pricing is a deterministic function of the trade size.

Hedging LPs

In order to hedge delta exposure for LPs, Lyra needs to be able get long and short exposure on the base asset of the market. Synthetix is the only protocol where this can be done in one place. This is because Synthetix has both the exchange contracts - which allow for conversion between Synths - and also the shorting mechanism, which allows for loans to be taken against Synths and short sold into sUSD.