There is a full vector field $ v = (v_ {1}, v_ {2}): D rightarrow mathbb {R} ^ 2 $, where $ D $ is an open subset of $ mathbb {R} ^ 2 $, $ v in C ^ 1 $, with the following properties:

1) say $ l: = {x_ {2} = 0 } $, we have $ D cap l neq emptyset $ and $ v_ {2} (x) leq 0 $ for all $ x in l cap D $ and there is a point $ x in l $ S t. $ v_ {2} (x) <0 $;

2) The whole $ {x in l cap D s.t. v_ {2} (x) = 0 } $ has an accumulation point $ A en l cap D $

3) There is a point $ x in D $ with $ x_ {2} <0 $ and a time $ t in mathbb {R} $ S t. $ phi ^ {t} (x) 2> 0 $ (i.e., the integral curve from $ x $ sometime it reaches half of the upper plane).

The motivation for the question lies in the following statement "if the vector field $ v $ never points to the middle of the top plane along $ l $, then each solution from the lower half plane remains in the closed lower half plane, "which I cannot say whether it is true or not. Actually, I know that if the vector field is tangent throughout $ l $ or if the set of points where it is tangent has no accumulation point in $ l $, then the claim is true. This motivates 1) and 2).

Finally let me say that I would take as an answer an example where $ v $ it is not complete.

Thank you for reading.