from too much hairline in the soup to equations:

\begin{eqnarray}

\frac{\partial T}{\partial x}&=&-\frac{ z}{ v}\frac{\partial \rho}{\partial t} \nonumber \\

\frac{\partial \rho}{\partial x}&=&-\frac{ 1}{ vz}\frac{\partial T}{\partial t} \nonumber \\

\end{eqnarray}

any insight? v velocity, t time etc. Similar to Maxwell?

there's no [$]z[$] derivatives? incorporate [$]z[$] into either [$]T[$] or [$]\rho[$], [$]\hat{\rho}=z\rho[$]

\begin{eqnarray}

\frac{\partial \hat{\rho}}{\partial t}+v\frac{\partial T}{\partial x}&=&0 \nonumber \\

\frac{\partial T}{\partial t} +v\frac{\partial\hat{\rho}}{\partial x}&=&0\nonumber \\

\end{eqnarray}

one solution would be [$]T=\hat{\rho}[$] with [$]\frac{\partial \hat{\rho}}{\partial t}+v\frac{\partial\hat{\rho}}{\partial x}=0[$] which is a convective derivative, [$]\hat{\rho}[$] (and [$]T[$]) are constant on a blob of material as it moves

For more general solutions,

if [$]v[$] depends on either [$]x[$] or [$]t[$], you need more information (3 unknowns, 2 equations)

if [$]v[$] is independent of both [$]x[$] and [$]t[$], eliminate [$]\hat{\rho}[$] or [$]T[$]

\begin{eqnarray}

\frac{\partial^2 \hat{\rho}}{\partial t^2}+v\frac{\partial^2 T}{\partial x\partial t}&=&0 \nonumber \\

\frac{\partial^2 T}{\partial t\partial x} +v\frac{\partial^2 \hat{\rho}}{\partial x^2}&=&0\nonumber \\

\end{eqnarray}

\begin{eqnarray}

\frac{\partial^2 \hat{\rho}}{\partial t^2}-v^2\frac{\partial^2 \hat{\rho}}{\partial x^2}&=&0 \nonumber \\

\frac{\partial^2 T}{\partial t^2} -v^2 \frac{\partial^2 T}{\partial x^2}&=&0\nonumber \\

\end{eqnarray}