  A Thurston map is called nearly Euclidean if its local degree at each
critical point is 2 and it has exactly four postcritical points. Nearly
Euclidean Thurston (NET) maps are simple generalizations of rational Lattes
maps. We investigate when such a map has the property that the associated
pullback map on Teichmuller space is constant. We also show that no Thurston
map of degree 2 has constant pullback map
