  Group classification of the three-dimensional equations describing flows of
fluids with internal inertia, where the potential function $W=
W(\rho,\dot{\rho})$, is presented. The given equations include such models as
the non-linear one-velocity model of a bubbly fluid with incompressible liquid
phase at small volume concentration of gas bubbles, and the dispersive shallow
water model. These models are obtained for special types of the function
$W(\rho,\dot{\rho})$. Group classification separates out the function
$W(\rho,\dot{\rho})$ at 15 different cases. Another part of the manuscript is
devoted to one class of partially invariant solutions. This solution is
constructed on the base of all rotations. In the gas dynamics such class of
solutions is called the Ovsyannikov vortex. Group classification of the system
of equations for invariant functions is obtained. Complete analysis of
invariant solutions for the special type of a potential function is given.
