  The Schr\"{o}dinger equation $\psi"(x)+\kappa^2 \psi(x)=0$ where
$\kappa^2=k^2-V(x)$ is rewritten as a more popular form of a second order
differential equation through taking a similarity transformation
$\psi(z)=\phi(z)u(z)$ with $z=z(x)$. The Schr\"{o}dinger invariant $I_{S}(x)$
can be calculated directly by the Schwarzian derivative $\{z, x\}$ and the
invariant $I(z)$ of the differential equation $u_{zz}+f(z)u_{z}+g(z)u=0$. We
find an important relation for moving particle as $\nabla^2=-I_{S}(x)$ and thus
explain the reason why the Schr\"{o}dinger invariant $I_{S}(x)$ keeps constant.
As an illustration, we take the typical Heun differential equation as an object
to construct a class of soluble potentials and generalize the previous results
through choosing different $\rho=z'(x)$ as before. We get a more general
solution $z(x)$ through integrating
$(z')^2=\alpha_{1}z^2+\beta_{1}z+\gamma_{1}$ directly and it includes all
possibilities for those parameters. Some particular cases are discussed in
detail.
