In a deep pot mix the fish head (see notes), olive oil, lime juice, garlic cloves, and half of the auyama. Pour in 2.5 quarts [2.5 liters] of water. Boil covered over medium-high heat until the auyama has dissolved and heads are cooked and flaky (25 to 35 minutes).Remove from the heat and sieve to discard the solids. You can return any large pieces of cheek meat to the broth if you wish to dig it out.
2. Cook the vegetables
Return the broth to the pot. Add cilantro, carrots, potato, and remaining auyama. Cook covered over medium-low heat until the vegetables are cooked through (about 10 minutes).
3. Add the fish
Add the fish fillet to the pot and cook until it is cooked through (5-10 minutes). In the end, you should have about half the liquid you started with, if it's too little, add some more water and heat through, if it's too much, simmer some more to reduce it. This is not a thick soup, the liquid has the consistency of broth.
4. Serve
Season with salt and pepper to taste. Remove from the heat and serve hot.
Video
Notes
My mom did not strain the soup as in step 1, but I hate to find bones in it, so here I turn away from mami's recipe.