SEEMOUS 2013 Problem 2

Let $M,N\in M_2(\mathbb C)$ be two nonzero matrices such that $$M^2=N^2=0_2\text{ and }MN+NM=I_2$$ where $0_2$ is the $2\times2$ zero matrix and $I_2$ the $2\times2$ unit matrix. Prove that there is an invertible matrix $A\in M_2(\mathbb C)$ such that $$M=A\begin{pmatrix}0&10&0\end{pmatrix}A^{-1}\text{ and }N=A\begin{pmatrix}0&01&0\end{pmatrix}A^{-1}.$$