  We study solutions of the equation $$ g_t-g_{tyy} + 4g^2 - 4gg_{yy} = y
gg_{yyy}-yg_yg_{yy}, \qquad y\in\mathbb{R},$$ which arises by considering
solutions of the Euler-Arnold equation on a contactomorphism group when the
stream function is of the form $f(t,x,y,z) = zg(t,y)$. The equation is
analogous to both the Camassa-Holm equation and the Proudman-Johnson equation.
We write the equation as an ODE in a Banach space to establish local existence,
and we describe conditions leading to global existence and conditions leading
to blowup in finite time.
