  The Euler-$\alpha$ equations model the averaged motion of an ideal
incompressible fluid when filtering over spatial scales smaller than $\alpha$.
We show that there exists $\beta>1$ such that weak solutions to the two and
three dimensional Euler-$\alpha$ equations in the class $C^0_t H^\beta_x$ are
not unique and may not conserve the Hamiltonian of the system, thus
demonstrating flexibility in this regularity class. The construction utilizes a
Nash-style intermittent convex integration scheme. We also formulate an
appropriate version of the Onsager conjecture for Euler-$\alpha$, postulating
that the threshold between rigidity and flexibility is the regularity class
$L^3_t B^{\frac{1}{3}}_{3,\infty,x}$.
