Assume that $M$ is a surface in $\mathbb{R}^{3}$. We denote its shape operator by $S$. A vector field $X$ is shape related to $Y$ if $S(X)=Y$. (of course it is not an equivalent relation).

Assume that $X, Y$ are two vector fields on $S^{2}$ with the same singular points. Is there a surface $M$, diffeomorphic to $S^{2}$, such that $X$ is shape related to $Y$.

Note that the initial vector fields $X$ and $Y$, are pull backed via diffeomorphism to two new vector fields on $M$. these 2 new vector fields are denoted by $X$ and $Y$, again

Assume that $M$ is a surface in $\mathbb{R}^{3}$. We denote its shape operator by $S$. A vector field $X$ is shape related to $Y$ if $S(X)=Y$. (of course it is not an equivalent relation).

Assume that $X, Y$ are two vector fields on $S^{2}$ with the same singular points. Is there a surface $M$, diffeomorphic to $S^{2}$ via a diffeomorphism $\phi: M \to S^{2}$, such that $\phi^{*} (X)$ is shape related to $\phi^{*}(Y)$.