There are many inverters (all-in-one or component systems) that will send a 2-wire signal to start a generator depending on battery SoC and sometimes time of day. Your biggest problem is too many choices.
I have SMA Sunny Island & Sunny Boy, but I use it as grid-backup. It could have had a generator for input instead.
Other top names to consider include Midnight, Schneider, Outback, Victron, SolArk.
There are other lower-cost brands, some sold and supported in the US, some shipped from overseas.
EG4, MPP, many others.
As B-Mod says, have to determine what you want it to do to size the system.
Batteries are the biggest deal. You may want to select a lithium battery with BMS that talks to inverter, although some systems operate open-loop, just based on voltage limits. Some people still use lead-acid. It is simpler, tolerant of cold, but less efficient, shorter lived unless oversize, and may cost more up front.
You would want usable battery capacity to at least last through one night. If you also have grid, that is likely plenty, maybe even get by with smaller battery. If no grid, you may want enough battery for 3 days without sun. Or, since you have generator, use that any time sun is insufficient.
PV itself is pretty cheap, so oversizing PV array minimizes use of generator, and can even keep battery full later in the day so a bit less capacity needed to make it through the night.
You can probably spend anywhere from $1500 to $15,000 or more, so important to figure out how much capacity you need and want. And chose between cheapest available vs. those with the best reputation. e.g. some people still use a Trace inverter system they bought 30 years ago (that product line has evolved into Schneider, and the engineers are now at Midnight.)
