This is a question related to A question about a one-form on Riemannian manifold
Let M be a Riemannian Manifold, how to construct two vector fields that they didn't have common zeros and perpendicular to each other with the same length only on isolated points. I mean two vector fields $X$ and $Y$, $zero(X)\cap zero(Y)=\varnothing$ and $\langle X(p),Y(p)\rangle=0$ and $|X|=|Y|$ only on isolated points if $X(p)\neq 0,Y(p)\neq 0$. I want to know how to construct the two vector fields?
Edit: We assume $X$ and $Y$ are two smooth vector fields. I don't know whether the vector fields exist on any Riemannian Manifold, maybe need some condition.
I am sorry, I lost a condition, I need they "perpendicular to each other with the same length only on isolated points", so they can perpendicular on a submanifold. But if they have the same length then only on isolated points.