IMC 2019 Problem 2

A four-digit number $YEAR$ is called very good if the system \begin{align*} Yx+Ey+Az+Rw& =Y Rx+Yy+Ez+Aw & = E
Ax+Ry+Yz+Ew & = A Ex+Ay+Rz+Yw &= R \end{align*} of linear equations in the variables $x,y,z$ and $w$ has at least two solutions. Find all very good $YEAR$ s in the 21st century.

(The $21$ st century starts in $2001$ and ends in $2100$ .)

Proposed by Tomáš Bárta, Charles University, Prague