  We investigate the time-dependent perturbations of strongly coupled
$\mathcal{N} = 4$ SYM theory at finite temperature and finite chemical
potential with a second order phase transition. This theory is modelled by a
top-down Einstein-Maxwell-dilaton description which is a consistent truncation
of the dimensional reduction of type IIB string theory on AdS$_5\times$S$^5$.
We focus on spin-1 and spin-2 sectors of perturbations and compute the
linearized hydrodynamic transport coefficients up to the third order in
gradient expansion. We also determine the radius of convergence of the
hydrodynamic mode in spin-1 sector and the lowest non-hydrodynamic modes in
spin-2 sector. Analytically, we find that all the hydrodynamic quantities have
the same critical exponent near the critical point $\theta = 1/2$. Moreover, we
establish a relation between symmetry enhancement of the underlying theory and
vanishing the only third order hydrodynamic transport coefficient $\theta_1$,
which appears in the shear dispersion relation of a conformal theory on a flat
background.
