How do you find the solutions (streamlines and integral surfaces) for $W(x,y,z)=\big(x\log x,-y\log y, -z\log z\big)$ for $0<x,y,z<1?$
I was able to plot the streamlines of $W:$
I can solve $W(x,y)$ exactly but am unsure how to approach the generalized problem in 3d.
With "streamlines" I suppose you mean a (maximal) solution of the equation $$\frac{d}{dt}(x(t),y(t),z(t))=W(x(t),y(t),z(t))\\=(W_x(x(t),y(t),z(t)),W_y(x(t),y(t),z(t)),W_z(x(t),y(t),z(t)))$$ or the set of all reparametrizations of that solution or the image of that solution. Well, the system of equations $$\frac{d}{dt}x(t)=W_x(x(t),y(t),z(t))=x(t)\log(x(t))\\ \frac{d}{dt}y(t)=W_y(x(t),y(t),z(t))=-y(t)\log(y(t))\\ \frac{d}{dt}z(t)=W_z(x(t),y(t),z(t))=-z(t)\log(z(t))$$ decouples and has the solution $(x(t),y(t),z(t))=(x(0)^{\exp(t)},y(0)^{\exp(-t)},z(0)^{\exp(-t)})$.
As for "integral surfaces", I suppose you mean a family of 2D surface which partition $\mathbb{R}^3$ and with each member of that family intersecting at 90° with each of the above integral curves. If we could find a pair of functions $f,g:\mathbb{R}^3 \to \mathbb{R}$ so that $W=f\nabla g$, then the contours of $g$, i.e. $\{x\in \mathbb{R}^3|\,g(x)=C\}_{C\in \mathbb{R}}$, would provide a formula for that family of surfaces. In this case we can just take $f\equiv 1$ and $$g(x,y,z):=\frac{1}{4}\left(x^2(2\log(x)-1)-y^2(2\log(y)-1)-z^2(2\log(z)-1)\right).$$ (I found $g$ by adding the anti-derivative of $W_x$ w.r.t. $x$, the anti-derivative of $W_y$ w.r.t. $y$ and the anti-derivative of $W_z$ w.r.t. $z$)