Ytukyg

Let $u$ be a harmonic function on a connected open set. If $int_{gamma}*du = 0$ for any cycle $gamma$ then $u$ has a harmonic function.

This question arises from an answer to this post

Please do modify this question's assumptions if I am not stating some things correctly.

We know that the converse is true, namely that for any harmonic $u$ on an open connected set that has a harnonic conjugate, $int_{gamma}*du = 0$. I am thinking maybe Morera's theorem has to do something with this..

I suggest looking at this from a real calculus of two variables perspective.
You are given that
$$int_gamma -frac{partial u}{partial y} mathrm{d} x+frac{partial u}{partial x} mathrm{d} y equiv 0, $$
for all cycles $gamma$ in the domain. This allows you to safely define a function
$$v(x,y) := int_{(x_0,y_0)}^{(x,y)} -frac{partial u}{partial y} mathrm{d} x+frac{partial u}{partial x} mathrm{d} y . $$ with $(x_0,y_0)$ a fixed point in the domain. The expression for $v(x,y)$ is well defined, since the difference between any two integration paths is a cycle.