  A 4-parameter polynomial family of equations generalizing the Camassa-Holm
and Novikov equations that describe breaking waves is introduced. A
classification of low-order conservation laws, peaked travelling wave
solutions, and Lie symmetries is presented for this family. These
classifications pick out a 1-parameter equation that has several interesting
features: it reduces to the Camassa-Holm and Novikov equations when the
polynomial has degree two and three; it has a conserved $H^1$ norm and it
possesses $N$-peakon solutions, when the polynomial has any degree; and it
exhibits wave-breaking for certain solutions describing collisions between
peakons and anti-peakons in the case $N=2$.
