DOS for anisotropic 2D electron gas dispersion

If we start with the simple 2D isotropic-parabolic dispersion,

begin{align}
Eleft(textbf{p}right) & approxtilde{varepsilon}_{0}+alpha p_{y}^{2}+alpha p_{x}^{2},
label{1}
end{align}
where $alpha>0$, the DOS can be obtained through
begin{align}
{cal D}left(Eright)=intfrac{dtextbf{p}}{left(2piright)^{2}}deltaleft[E-Eleft(textbf{k}right)right] & =intfrac{dtextbf{p}}{left(2piright)^{2}}deltaleft[tilde{varepsilon}_{0}+alpha p_{y}^{2}+alpha p_{x}^{2}-Eright]nonumber
& =frac{1}{left(2piright)^{2}}int dp_{x}dp_{y}deltaleft[tilde{varepsilon}_{0}+alphaleft(p_{x}^{2}+p_{y}^{2}right)-Eright],
end{align}
Using polar coordinates $p_{x}=pcostheta$, $p_{y}=psintheta$,
we obtain

begin{align*}
int dtextbf{p}deltaleft[E-Eleft(textbf{k}right)right] & =intint dtheta dppdeltaleft[tilde{varepsilon}_{0}-E+alpha p^{2}right]
& =2piint dpthinspace pthinspacedeltaleft[tilde{varepsilon}_{0}+alpha p^{2}-Eright]
end{align*}
Now we can use the delta-Dirac properties, yielding

begin{align}
int dppdeltaleft[tilde{varepsilon}_{0}+alpha p^{2}-Eright] & =int dppfrac{deltaleft(p-p_{1}right)}{left|frac{dleft(tilde{varepsilon}_{0}+alpha p^{2}-Eright)}{dp}right|_{p=p_{1}}}
& =int dppfrac{deltaleft(p-p_{1}right)}{2left|alpha p_{1}right|}
& =frac{p_{1}}{2left|alpha p_{1}right|}
& =frac{1}{2alpha}
end{align}
with (assumed $p>0$) $p_{1}=sqrt{frac{E-tilde{varepsilon}_{0}}{alpha}}$. Finally, we have
begin{align*}
{cal D}left(Eright) & =frac{1}{left(2piright)^{2}}times2pitimesfrac{1}{2alpha}
& =frac{1}{4pialpha},
end{align*}
which agrees with the literature result.

Now I’m trying to do the same calculation, but for the anisotropic case, i.e.,

begin{align}
Eleft(textbf{p}right) = tilde{varepsilon}_{0}-alpha p_{y}^{2}+beta p_{x}^{2},
end{align}
with $alpha,beta>0$.

Using polar coordinates $p_{x}=pcostheta$, $p_{y}=psintheta$,here we obtain
begin{align*}
int dtextbf{p}deltaleft[E-Eleft(textbf{k}right)right] & =intint dtheta dppdeltaleft[tilde{varepsilon}_{0}-E-alpha p^{2}sin^{2}theta+beta p^{2}cos^{2}thetaright]
& =intint dtheta dppdeltaleft[tilde{varepsilon}_{0}-E-left(alpha+betaright)p^{2}sin^{2}theta+beta p^{2}right]
end{align*}
Using again the delta-Dirac properties, we obtain
begin{align}
int dppdeltaleft[tilde{varepsilon}_{0}-E-alpha p^{2}sin^{2}theta+beta p^{2}cos^{2}thetaright] & =int dppfrac{deltaleft(p-p_{1}right)}{left|frac{dleft(tilde{varepsilon}_{0}-E-left(alpha+betaright)p^{2}sin^{2}theta+beta p^{2}right)}{dp}right|_{p=p_{1}}}
& =int dppfrac{deltaleft(p-p_{1}right)}{2left|-left(alpha+betaright)p_{1}sin^{2}theta+beta p_{1}right|}
& =frac{p_{1}}{2left|-left(alpha+betaright)p_{1}sin^{2}theta+beta p_{1}right|}
& =frac{1}{2left|-left(alpha+betaright)sin^{2}theta+betaright|}
end{align}
with $p_{1}=sqrt{frac{tilde{varepsilon}_{0}-E}{left(alpha+betaright)sin^{2}theta-beta}}$. Finally, we can write
begin{align}
{cal D}left(Eright)=int_{0}^{2pi}dthetafrac{{rm 1}}{2left|beta-left(alpha+betaright)sin^{2}thetaright|}
end{align}

Although it seems that I cannot solve this integral analytically, what is concerning me is another aspect, more specifically, that this dispersion should contain DOS van Hove singularity at $E = tilde{varepsilon_0}$ $({cal D}(E)approx ln|E- tilde{varepsilon_0}|)$, and I’m not obtaining nothing similar.