What economic systems will lead to a stable, normal distribution?

Under capitalism, wealth will eventually accrue to the wealthy, leaving the poor very poor.

Socialism with a provision for capitalism has a chance, as it can provide all with the basics, and allow some to become wealthy while preventing the very poor from having to do without essentials.

Pure socialism, wherein the basic industries and production from the land are owned by the State and used for the benefit of the general public might be the best. It's positively criminal, for instance, for oil to be pumped from under US soil for the benefit of Big Oil. That should be funding the government.

And it's criminal for the defense budget and military to be spent protecting Big Oil.