  We construct generating technique for 5D minimal and $U(1)^3$ supergravities
based on hidden symmetries arising in dimensional reduction to three
dimensions. In the three-vector case the symmetry is SO(4,4), and the minimal
case corresponds to contraction of this group to $G_{2(2)}$. The matrix
representation is presented applicable to both cases and the generating
transformations preserving an asymptotic structure are listed. Our
transformations contain enough free parameters to construct the general charged
black ring in $U(1)^3$ theory starting with known solutions. To avoid a
complicated inverse dualisation in the component form we introduce the
matrix-valued dualisation which opens the way to derive new solutions purely
algebraically from the coset representation of the seed.
