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Your last equation $$\mathcal K=v_t-xv_z-v_{xx}=0 \tag 1$$ is indeed a particular case of Hörmander's $X_0-\sum_{1\le j\le r}X_j^2$ with $$X_0=\partial_t-x\partial_z, X_1=\partial_x, r=1, [X_0,X_1]=\partial_z.$$ However, it is also exactly Kolmogorov equation, as studied by Andrei Kolmogorov in his 1937 Annals paper. This article was in fact the starting point of Lars Hörmander's work on this topic. It turns out that there is an explicit parametric construction for (1): a change of variables straightening the vector field $X_0$ is $$\begin{cases} s=t,\ \ x_1=x,\ \ x_2=z+xt, \end{cases}$$ so that $\mathcal K=\partial_s-(\partial_{x_1}+s\partial_{x_2})^2$ and the latter can be Fourier transformed to the ODE $$\partial_s+(\xi_1+s\xi_2)^2.$$ The latter is of course explicitly solvable: we have an explicit integral expression $$v(t,x,z)=v(s,x_1,x_2-sx_1)=w(s,x_1,x_2)=\iint e^{i(x_1\xi_1+x_2\xi_2)}e^{-\int_0^t(\xi_1+s\xi_2)^2ds} \hat w_0(\xi_1,\xi_2) d\xi_1d\xi_2,$$ $$v(t,x,z)=\iint e^{i(x\xi_1+(z+xt)\xi_2)}e^{-\int_0^t(\xi_1+s\xi_2)^2ds} \hat v_0(\xi_1,\xi_2) d\xi_1d\xi_2,\quad v_0(x,z)=v(0,x,z).$$

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Your last equation $$\mathcal K=v_t-xv_z-v_{xx}=0 \tag 1$$ is indeed a particular case of Hörmander's $X_0-\sum_{1\le j\le r}X_j^2$ with $$X_0=\partial_t-x\partial_z, X_1=\partial_x, r=1, [X_0,X_1]=\partial_z.$$ However, it is also exactly Kolmogorov equation, as studied by Andrei Kolmogorov in his 1937 Annals paper. This article was in fact the starting point of Lars Hörmander's work on this topic. It turns out that there is an explicit parametric construction for (1): a change of variables straightening the vector field $X_0$ is $$\begin{cases} s=t,\ \ x_1=x,\ \ x_2=z+xt, \end{cases}$$ so that $\mathcal K=\partial_s-(\partial_{x_1}+s\partial_{x_2})^2$ and the latter can be Fourier transformed to the ODE $$\partial_s+(\xi_1+s\xi_2)^2.$$ The latter is of course explicitly solvable.