What might the energy partitioning be, when the equipartition theorem is not applicable?

Question background. There is an equipartition theorem, and it is without doubt correct. But it has its conditions of applicability, which are not always satisfied. There are well-known examples of a chain of connected oscillators, the spectral density of a black body, the new example of an ideal gas in a round vessel. How may or may not the energy be partitioned in such cases, when the equipartition theorem is not applicable? Can anyone provide more systems with known uneven laws of energy partitioning?