I'm having hard time understanding the concept of symmetry in vector fields. The following is a paragraph from the book Geometric control theory by Jurdjevic.
"A vector field $X$ is said to be a symmetry of another vector field $Y$ if for any integral curve $\sigma(t)$ of $Y$, $(exp\lambda t)(\sigma(t))$ is also an integral curve of $X$ for each value of $\lambda$. Stated in slightly different language, a symmetry preserves the solution curves of $Y$, or the flow of $X$ permutes the integral curves of $Y$."
What does it means that flow of $X$ permutes the integral curves of $Y$? Does this means that flow of $X$ maps each integral curve to some other integral curve?
"...Expressing the condition of symmetry in terms of exponential maps yields
$(exp\lambda X)((exptY)(x))=(exptY)((exp\lambda X)(x)).$"
why does this follows from previous paragraph?
I get $(exp\lambda X)((exptY)(x))$ is an integral curve for $X$ but I can't see how to go further
I think there are some errors in the text you cited from the book: What it should say that for an integral curve $t\mapsto\sigma(t)$ of $Y$ and any fixed $\lambda\in\mathbb R$, the curve $t\mapsto (exp\lambda X)(\sigma(t))$ is an integral curve of $Y$. This means that any flow of $X$ maps each integral curve of $Y$ to an integral curve of $Y$, so it permutes the integral curves of $Y$.
Now if for the above integral curve $\sigma$, you put $x=\sigma(0)$, then $\sigma(t)=exp(tY)(x)$. So by the above, $t\mapsto exp(\lambda X)(exp(tY)(x))$ is an integral curve of $Y$, and for $t=0$, one gets the point $exp(\lambda X)(x)$. Uniqueness of integral curves then tells you that $exp(\lambda X)(exp(tY)(x))=exp(tY)(exp(\lambda X)(x)$, or that $exp(tY)\circ exp(\lambda X)=exp(\lambda X)\circ exp(tY)$.