  At a very low temperature of 9mK, electrons in the 2nd Landau level of an
extremely high mobility two-dimensional electron system exhibit a very complex
electronic behavior. With varying filling factor, quantum liquids of different
origins compete with several insulating phases leading to an irregular pattern
in the transport parameters. We observe a fully developed $\nu=2+2/5$ state
separated from the even-denominator $\nu=2+1/2$ state by an insulating phase
and a $\nu=2+2/7$ and $\nu=2+1/5$ state surrounded by such phases. A developing
plateau at $\nu=2+3/8$ points to the existence of other even-denominator
states.
