If you want to make the corporations truly public. Do an IPO of your EXISTING shares asap (the max first off offer is 15mil), now set the portfolio target for your country to whatever figure you want below 25% (I set it to 24.97%), then in the same month have your enterprise buy 23mil shares (if your order 24mil then you wont be able to set a portfolio target for enterprise between 24% and 25%, it just saves more hassle), (repeat if you have another enterprise) you will need to wait for your enterprise/s to gain ownership of some share to set the portfolio target. I'd recommend setting the enterprise/s portfolio target to 24.98-24.99%, and country target to .01% less then the enterprise target, this allows you to maintain maximum ownership of the corporation while keeping it truly public and making an enterprise the largest minority shareholder give the production tax income to your country. You don't have to do it that way, you could always create an IPO any time and set the country portfolio target to any figure under 25% but it makes your country less revenue and you have less security of ownership.
If you have an already public corporation thats up shit creek and wishing you could put some cash into it, you can in a tedious way. Hoping that you still can make IPOs in this corporations create a IPO with NEW shares then have your country/enterprise buy these new shares (because odds are the investment funds won't buy them) in result transferring your countries/enterprises cash into the corporation, the buying of these shares will create an artificial demand (speculation) increasing the price, which will probably allow you to create another IPO if needed. If you can't create an IPO you can probably buy shares that will be in high supply, doing this will hopefully allow you to create an IPO.
I don't know why you would want to create an IPO for Adams reason but his advise is the best for that result. If you want to sell off the corporation to gamemaster enterprise, then wait for the market value to become very high then simply create and IPO and set the portfolio target to 0%. This will slowly sell off shares creating a high return but probably not as high as possible if you were to micromanage.
If you don't know or have a reason to create and IPO then don't, they are a lot of hassle especially when you don't have a reason.