How to find compatible motherboard for the computer

What you're experiencing here is a conflict between case and mainboard design. Both the case and the mainboard may conform to the full- or mid- ATX specification, but within that specification there is not usually requirements as to where connectors go on the board, or where bays are arranged in the case or chassis.

This is just something you have to deal with and be careful of, there is not another specification or label that will guarantee better fit.

Prior examining and comparison of the desired mainboard and case are the only way around this issue.

It seems to me the CC515 is a mid-size ATX case rather than a full-size. But that difference is not generally a truly different specification, just a different size case.

I believe the only different ATX specs are Micro ATX and ATX (or full ATX). Micro ATX cases and boards are very small and usually found in very specialized systems such as those made by Shuttle.

In all likelihood, if the system uses a proprietary form-factor, it probably won't be the actual physical dimensions (width, length) that are the problem but rather the placement of components on the backplane and of the standoff holes.

If the backplane is unusual, you can probably/hopefully just pop in a compatible bezel (figure 1), though the case may not be designed to support replacing the bezel (you may be able to just cut a hole).

If the standoff holes (figure 2) are in the wrong place, then you may have to find other means of securing the board to the case, or if you are lucky, perhaps secure it with just one or two screws.

FormFactors.org has a comprehensive list of form-factors for motherboards, power-supplies, and other components, with links to PDFs of the specifications for each that include dimensions and mounting-hole placements.

Figure 1: Variety of bezels:

Figure 2: Standoff holes: