Let's the value be m and its shifted version n.
2/3*n = m
m and n are both integers, so n-m=1/3*n is also an integer.
Therefore 3 | n .
As their digits have the same sum, 3 | m.
Therefore 3 | n-m --> 3 | 1/3*n --> 9 | n.
As their digits have the same sum, 9 | m.
Following similar logic from above, above 9 | n-m --> 9 | 1/3*n --> 27 | n.
So you need only account for 1/27 of the integers for a brute strength solution (which speeds it up significantly)
I got 285714 and 571428.