  By generalizing Bo Gao's approach [Phys. Rev. A 58, 1728 (1998)] for solving
the Schr\"{o}dinger equation for an isotropic van der Waals (vdW) potential to
the systems with a multi-scale anisotropic long-range interaction, we derive
the solutions for the Schr\"{o}dinger equation for an anisotropic dipole-dipole
interaction plus an isotropic attractive vdW potential, i.e.,
${C_d(1-3\cos^2\theta)}/{r^3}-{C_6}/{r^6}$, which is projected to the subspace
with angular momentum $l\leq l_{\rm cut}$, with $l_{\rm cut}$ being an
arbitrary angular-momentum cutoff. Here $\theta$ is the polar angle of the
coordinate $\boldsymbol{r}$ and $r=|\boldsymbol{r}|$. The asymptotic behaviors
of these solutions for $r\rightarrow 0$ and $r\rightarrow \infty$ are obtained.
These results can be used in the research of collisions and chemical reactions
between ultra-cold polar molecules in a static electric field. Our approach to
derive the solutions can be applied to the systems with a general long-range
potential $\sum_{\lambda= 2}^{\lambda_{\rm max}}
{V_\lambda(\theta,\varphi)}/{r^\lambda}$, with $\varphi$ being the azimuthal
angle of $\boldsymbol{r}$, and thus can be used in various problems on
molecule-molecule interaction.
