Are you familiar with what "private equity" is? Note that "private equity" is distinct from "private" "equity."
Private equity is an asset class. equity securities and debt (such as, oh, unpaid medical bills, and the right to collect on them) can be purchased as well as corporations/corporate debt.
Are you sure you understand what I'm saying?
Second, what exactly do you accomplish by doing this, except paying a bunch of money in incorporation fees and drafting documents?
There's a reason why corporate shell games are usually played by people with relatively deep pockets: all the benefits that come from using corporate law require lawyers and accountants, and those are expensive.