Show that z-axis of Lorenz equations is invariant

Question

Given Lorenz equations \begin{align} \dot{x} &= \sigma(y-x)\\ \dot{y} &= rx - y- xz\\ \dot{z} &= xy - bz \end{align} suppose $x(0)=0=y(0)$.
Prove that $x(t)=0=y(t)$ for all $t \ge 0$.

Intuitively, I can see that this is true. But I am not sure how to give a rigorous proof.
Could you give some hints or references to related concepts?

Answer 1

$(x(t),y(t),z(t))=(0,0,z_0 e^{-bt})$ is a solution with initial data $(0,0,z_0)$, and the solution to the initial value problem is unique by the Picard–Lindelöf theorem.

More conceptually, on the $z$ axis the vector field equals $(0,0,-bz)$, which is tangent to the $z$-axis. Therefore the $z$ axis is invariant.