On conditions for the zero regression of one linear statistic with respect to another

Let $X_1$, $X_2$ be independent identically distributed random vectors in $R^2$; $A_1$, $A_2$, $B_1$, $B_2$ be non-singular $(2\times2)$ matrices, $Y_1=A_1X_1+A_2X_2$, $Y_2=B_1X_1+B_2X_2$. The condition $\mathbf E(Y_1\mid Y_2)=0$ is studied.