I've already generated my first and collected a few more, but the game won't let me generate another syvesti. Is there a limit or something I'm doing wrong? (Sorry if this sounds a little dumb or I didn't read the rules properly. I'm pretty new here.)

The first pet you generate was a freebie. Now if you want to generate a pet you need an essence of the specific species you are interested in. Essences are generally pretty expensive so you'll need to save up, or get lucky and find one nurturing or in mission satchels (both are really rare).

I very rarely use essences and hoard them for newer traits that are harder to find pets up for breeding with.

If you want more pets you have to buy, win, catch, or be gifted one. Before you know it you'll be overflowing with pets!

For some extra info on Essences you can look at this previous FAQ post we answered on the topic
Essences + ways to obtain them + what the differences between the three types are

(after obtaining an essence, if you wish to use it, you go to the generator and click on the 'load essence' text, select the rarity of the essence, then scroll, click once you find it, and hit 'load essence'. Switching essence to a different species will clear traits but retain colours, and switching rarity level of a single species will retain all traits and colours)

(Accidental post)

Ah, okay. Thank you all!