just economic interpretation: Is every steady state a golden rule steady state? Is every steady state a golden rule steady state in solow model?

I know this question’s exact answer. I only want to ask for the following,

When know that at steady state $\dot{k}=0$ 

And in solow swan model $$\dot{k}= sk_t^a-(\delta + n+g)k_t$$

Since this equation is zero, I obtain the $$k_{ss}=(\frac{s}{\delta + n+ g})^{1/1-a}$$

When I maximize $$c_{ss}=f(k_{ss})-(\delta + n+g)k_{ss}$$ with respect to $k_{ss}$

I obtain the golden rule level $k_{gr}=(\frac{a}{\delta + n+ g})^{1/1-a}$

So, 

$$k_{gr}=k_{ss} \iff a=s$$

That’s, saving rate is equal to capital share of income. 

Otherwise, under different conditions, they are not the same.

What i want to ask that to what extent such a expression in the view of an economist is logical? This is true in terms of mathematical expression, but is it logical in terms of economic intuition  as well? If correct, What does this result say to me intuitively?
No. There are many steady state levels of per-capita consumption. The maximum of these is then defined to be the golden rule (steady state level of per-capita consumption).