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In this paper, we discuss the regularities of energy of each angular momentum$I$ averaged over all the states for a fixed angular momentum (denoted as$\bar{E}_I$'s) in many-body systems interacting via a two-body random ensemble.It is found that $\bar{E}_I$'s with $I \sim I_{min}$ (minimum of $I$) or$I_{max}$ have large probabilities (denoted as ${\cal P}(I)$) to be the lowest,and that ${\cal P}(I)$ is close to zero elsewhere. A simple argument based onthe randomness of the two-particle cfp's is given. A compact trajectory of theenergy $\bar{E}_I$ vs. $I(I+1)$ is found to be robust. Regular fluctuations ofthe $P(I)$ (the probability of finding $I$ to be the ground state) and ${\calP}(I)$ of even fermions in a single-$j$ shell and boson systems are found to bereverse, and argued by the dimension fluctuation of the model space. Otherregularities, such as why there are 2 or 3 sizable ${\cal P}(I)$'s with $I\simI_{min}$ and ${\cal P}(I) \ll {\cal P}(I_{max})$'s with $I\sim I_{max}$, whythe coefficients $C$ defined by $<\bar{E}_I >$=$CI(I+1)$ is sensitive to theorbits and not sensitive to particle number, are found and studied for thefirst time.Comment: 19 pages and 6 figure

Many-body systems interacting via a two-body random ensemble. II. Average energy of each angular momentum