  Axial algebras are a class of non-associative algebra with a strong link to
finite (especially simple) groups which have recently received much attention.
Of primary interest are the axial algebras of Monster type $(\alpha, \beta)$,
of which the Griess algebra (with the Monster as its automorphism group) is an
important motivating example. In this paper, we complete the classification of
the symmetric $2$-generated primitive axial algebras of Monster type $(\alpha,
\beta)$.
  By previous work of Yabe, and Franchi and Mainardis, any such algebra is
either explicitly known, or is a quotient of the infinite-dimensional Highwater
algebra $\mathcal{H}$, or its characteristic $5$ cover $\hat{\mathcal{H}}$. In
this paper, we classify the ideals of $\mathcal{H}$ and $\hat{\mathcal{H}}$ and
thus their quotients. Moreover, we give explicit bases for the ideals. In fact,
we proceed in a unified way, by defining a cover $\hat{\mathcal{H}}$ of
$\mathcal{H}$ in all characteristics and classifying its ideals. Our new
algebra $\hat{\mathcal{H}}$ has a previously unseen fusion law and provides an
insight into why the Highwater algebra has a cover which is of Monster type
only in characteristic $5$.
