Definition of steady Ricci soliton Firstly, my English is poor, so I am not sure that which  is solution , and the steady solution  is the solution of evolution equation  in picture below ?
Secondly, what is the symmetry group of a equation ? As I know , the symmetry group is the all permutations on some symbols. But,seemly, there are nothing between equation and permutation. (the equation in the picture below is Ricci flow i.e. $\partial_tg_{ij}=-2R_{ij}$).
Thirdly , how to get a 1-parameter subgroup ? As I know ,the 1-parameter group is a family $(\varphi_t)_{t\in I}$ ($I$ open interval with $0\in I$) of diffeomorphisms from $M$  to $M$ and satisfy 
$$
\varphi_t\circ\varphi_s(q)=\varphi_{t+s}(q)~~~~if~~s,t,s+t\in I_q
$$ 
In fact ,I have suspicion about that  when $s,t$ is large enough , $s+t\notin I$. So ,it is not a group.
Fourthly , what is the mean of move ? It means that if $g(x,t)$ is solution of equation ,then $g(x,t)\circ\varphi_s$ still be  solution of equation?



 A: If $\phi(t) : M\rightarrow M $ is a family of diffeomorphism then
assume that $$ g(t)=\phi(t)^\ast g(0),\ \frac{\partial }{
\partial t} g(t)=-2 {\rm Ric} (g(t)) $$
That is $g(t)$ is steady soliton in general sense
If $X(x):=\frac{\partial }{\partial t}\bigg|_{t=0}\phi(t,x)$, then
$$ L_X g(0)=\frac{\partial }{
\partial t} g(t) =-2 {\rm Ric} (g(0))
$$
So $g(0)$ is steady Ricci solition if $L_X g(0)=-2 {\rm Ric} (g(0))$
From this, steady soliton $\Rightarrow$ steady Ricci solition Now we
will prove opposite direction :
If $\phi (t)$ is flow of $X$ define $$g(t,x):=\phi(t)^\ast g(0)
(\phi(t,x))$$ Then
\begin{align*}
\frac{\partial }{\partial t}\bigg|_{t=t_0} g(t,x)&=\frac{\partial
}{\partial s}\bigg|_{s=0} g(t_0+s,x)\\&= \frac{\partial }{\partial
s}\bigg|_{s=0} \phi(s+t_0)^\ast g(0)(\phi(s+t_0,x))
\\&=\frac{\partial
}{\partial s}\bigg|_{s=0} \phi(t_0)^\ast \phi(s)^\ast g(0)\bigg(\phi
(s, \phi(t_0,x))\bigg)
\\&=\frac{\partial
}{\partial s}\bigg|_{s=0} \phi(t_0)^\ast
 g(s)( \phi(t_0,x))
\\&=  \phi(t_0)^\ast \bigg(-2{\rm Ric}\ ( g(0) (\phi(t_0,x)) )
\bigg)
\\&= -2{\rm Ric}\ (\phi(t_0)^\ast g(0))
\end{align*}
Hence we complete the proof
