Reflection of singularities of solutions to systems of differential equations

where P EPS(0) is a pseudo-differential operator of order zero. We make the assumption that P(y,x,q,t)=det(qiG, (y ,x , t ) ) is real and has simple characteristics. Then, as is well known (see [ 1 I), singularities of solutions to (1.1) propagate along the null bicharacteristic strips of p in the interior of 9. Actually, the reference does not quite apply, since a/ayG is not a pseudodifferential operator on 9 (see the appendix). Suppose ( x o , t o ) E T*( a s2) 0 and that j null-bicharacteristic strips of p pass over (xo,to). That means there a r e j real solutions q,,. . . ,q, of p ( O , ~ ~ , q , [ ~ ) = 0 . The associated bicharacteristics y, ( t) = (y (t), x ( t ) , (t), t( t)) solve the equations