Camaro still dead?

Good evening I finally had my camaro hauled to me yesterday by towing service (which the ADAC kindly paid for for what I was very grateful for). Now it is at least in the dry and I can now begin to limit the error, find and fix. The first thing I did yesterday was to recharge the battery again because it was already a bit empty due to the long standing time in the workshop. This then lasted until today. Well in the time I leave the whole day here in the forum and elsewhere in the net. Printed me directions etc. Now I thought before I wanted to recharge the battery but you try to start it again. And look there he jumped on for exactly half a minute. I could even put in the 1speed and start it. Then he went out. My suspected is now really that it has something to do with the ignition system??? Because if e.g. the control chain would have skipped then he would not even start again or??? if the battery would have been broken then it would not have been addressed any more. Further questions are: Where is the fuel filter?? I just want to exchange it at the opportunity. Is only the ECM and the cooling water sensor responsible for the mixing or the injection quantity???