Lightroom 5 on Mac OS Sierra?
Color management is broken in any Lr version earlier than 6/CC under Sierra. You get clipped shadows and heavy color banding. Apple dropped support for some API/framework used by Lightroom prior to 6/CC.
If that's not it, please provide a lot more detail.
Exactly, LR5 running on Sierra
I was searching a lot for solution, but unsuccessfully. Do you have any ideas how can I fix it? Or where can I find that API/framework?
You just have to bite the bullet and upgrade Lightroom to 6 (which works fine). There's nothing you can fix. These things are deeply integrated in the operating system.
This is Apple, not Lightroom. Apple's policy is that the operating system itself is directly involved in all color management, and if the OS changes, applications have to change with it, or they get left behind like in this case.
Windows has a different policy and this can't happen. Windows doesn't do any color management at all - it just hands everything over to the application itself. If the application color management works, it works. Simple, but efficient and reliable.