We review the continuous symmetry approach and apply it to find the solution, via the construction of constants of motion and infinitesimal symmetries, of the 3D Euler fluid equations in several instances of interest, without recourse to Noether's theorem. We show that the vorticity field is a symmetry of the flow and therefore one can construct a Lie algebra of symmetries if the flow admits another symmetry. 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 the flow admits extra symmetries. 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.