The examples provided by Sin in their article Complications with Stochastic Volatility Models might help to answer your questions.

I'm transcribing the abstract below:

We show a class of stochastic volatility price models for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what it is usually assumed in the finance literature. We also show the existence of martingale measures, however, and give explicit examples.

And this technical article covers general integral tests for the existence and non-existence of EMM and ELMM (e.g., Theorem 3.1).

The examples provided by Sin in their article Complications with Stochastic Volatility Models might help to answer your questions.

I'm transcribing the abstract below:

We show a class of stochastic volatility price models for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what it is usually assumed in the finance literature. We also show the existence of martingale measures, however, and give explicit examples.

And this technical article (No arbitrage in continuous financial markets, by Criens) covers general integral tests for the existence and non-existence of EMM and ELMM (e.g., Theorem 3.1).