  Let $(X, D)$ be a log variety with an effective holomorphic torus action, and
$\Theta$ be a closed positive $(1,1)$-current. For any smooth positive function
$g$ defined on the moment polytope of the torus action, we study the
Monge-Amp\`{e}re equations that correspond to generalized and twisted
K\"{a}hler-Ricci $g$-solitons. We prove a version of Yau-Tian-Donaldson (YTD)
conjecture for these general equations, showing that the existence of solutions
is always equivalent to an equivariantly uniform $\Theta$-twisted
$g$-Ding-stability. When $\Theta$ is a current associated to a torus invariant
linear system, we further show that equivariant special test configurations
suffice for testing the stability. Our results allow arbitrary klt
singularities and generalize most of previous results on (uniform) YTD
conjecture for (twisted) K\"{a}hler-Ricci/Mabuchi solitons or
K\"{a}hler-Einstein metrics.
