  We review and apply the continuous symmetry approach to find the solution of
the 3D Euler fluid equations in several instances of interest, via the
construction of constants of motion and infinitesimal symmetries, without
recourse to Noether's theorem. We show that the vorticity field is a symmetry
of the flow, so if the flow admits another symmetry then a Lie algebra of new
symmetries can be constructed. For steady Euler flows this leads directly to
the distinction of (non-)Beltrami flows: an example is given where the topology
of the spatial manifold determines whether extra symmetries can be constructed.
Next, we study the stagnation-point-type exact solution of the 3D Euler fluid
equations introduced by Gibbon et al. (Physica D, vol.132, 1999, pp.497-510)
along with a one-parameter generalisation of it introduced by Mulungye et al.
(J. Fluid Mech., vol.771, 2015, pp.468-502). Applying the symmetry approach to
these models allows for the explicit integration of the fields along pathlines,
revealing a fine structure of blowup for the vorticity, its stretching rate,
and the back-to-labels map, depending on the value of the free parameter and on
the initial conditions. Finally, we produce explicit blowup exponents and
prefactors for a generic type of initial conditions.
