Curves on $SU(4)$ whose adjoint action on $\mathfrak{su}(4)$ integrates to $0$ Given $\xi \in \mathfrak{su}(4)$ and positive $T \in \mathbb{R}$, is it possible to find all smooth curves $U_s \in SU(4)$ with $U_0 = I$ such that
$$\int_0^T U_s \xi U_s^{\dagger} ds =0\; ?$$
 A: It is not clear what you mean by 'find' (give an explicit parametrization?) and 'curve' (continuous? differentiable? smooth? closed?).  Certainly, it is possible to 'write down' many curves satisfying your requirements, but 'find'?  I don't know what that means.
The first thing to recognize is that the orbit 
$$
M_\xi = \mathrm{SU}(4)\cdot \xi = \left\{U\xi U^\dagger\ | U\in\mathrm{SU}(4)\right\}
\subset {\frak{su}}(4)\simeq \mathbb{R}^{15}
$$
is a smooth submanifold endowed with a natural $\mathrm{SU}(4)$-invariant measure with respect to which the average of the position vector in ${\frak{su}}(4)$ is zero.  This means, in particular, that the convex hull of $M_\xi$ contains $0$.  Consequently, there are many curves $\eta:[0,T]:M_\xi$ with $\eta(0)=\xi$ such that 
$$
\int_0^T \eta(s)\ \mathrm{d} s = 0
$$
and any such curve $\eta$ can be lifted to a curve in $\mathrm{SU}(4)$ since the natural mapping $\mathrm{SU}(4)\to M_\xi$ is a smooth fiber bundle with fibers being the left cosets of the subgroup $G_\xi$ consisting of those $g\in \mathrm{SU}(4)$ that satisfy $g\xi g^\dagger = \xi$.  It doesn't matter how you lift such an $\eta$, so the problem really is to describe the 'balanced' curves in $M_\xi$ that start at $\xi$.
Now, the balancing condition is only $15$ equations on $\eta$ and the set of (continuous, differentiable, smooth, or real-analytic; pick your flavor) curves $\eta:[0,T]\to M_\xi$ with $\eta(0)=\xi$ is infinite dimensional when you put any reasonable topology on it, so you are asking for a description of an infinite dimensional space.  This raises the question of how you are going to 'parametrize' this space in such a way that you could pick out the 15 parameters that need to vanish in order to lie in the 'balanced' set.  I don't see any particularly good way to do this, though I can see some ideas about computing the 'tangent space' if the space of curves is sufficiently nice and is given the right topology.
Here is a simpler problem; if you can tell us what you would take for an answer for it, maybe we can give you a similar answer for your question:  Consider the set of smooth curves $\gamma:[0,1]\to S^2\subset\mathbb{R}^3$ with $\gamma(0)=(0,0,1)$. Describe the codimension $3$ subset that consists of those $\gamma$ satisfying
$$
\int_0^1 \gamma(s)\ \mathrm{d} s = 0\in\mathbb{R}^3.
$$
