= Solution
Let $q_0$ be the <stationary point> and let $H=D^2\beta(q_0)$ be its real symmetric <Hessian matrix>. First suppose the <stationary point> is interior and nondegenerate, the <oscillatory integral amplitude> and <oscillatory integral phase> are sufficiently smooth, and boundary contributions are absent or smaller than the stationary contribution. Diagonalize $H$ by an orthogonal change of coordinates, with <eigenvalues> $h_1,h_2$. Locally,
$$
\beta(q_0+\xi)=\beta(q_0)+\tfrac12\xi^TH\xi+O(|\xi|^3).
$$
Scaling each principal coordinate by $\lambda^{-1/2}$ and evaluating its <Fresnel integral> gives the <two-dimensional stationary-phase formula>:
$$
\boxed{h(\lambda)\sim\frac{2\pi}{\lambda}
\frac{\alpha(q_0)}{\sqrt{|\det H|}}
\exp\!\left(i\lambda\beta(q_0)+\frac{i\pi}4\operatorname{sgn}H\right).}
$$
Here $\operatorname{sgn}H$ is the number of positive <eigenvalues> minus the number of negative ones: $2$, $0$ or $-2$. The formula gives the leading nonzero term when $\alpha(q_0)\ne0$. A <partition of unity> isolates this neighbourhood; <integration by parts> away from <stationary points> makes the remaining interior contribution smaller. For a smooth compactly supported <oscillatory integral amplitude> the local error is $O(\lambda^{-2})$.
The printed assumption of a single <stationary point> alone is insufficient for an unconditional answer. A degenerate <oscillatory integral phase> $\beta=x^4+y^2$ with a smooth <oscillatory integral amplitude> supported near the origin has only one <stationary point>, but scales as $\lambda^{-3/4}$. Boundary terms can also contribute at order $\lambda^{-1}$: on the unit disk with <oscillatory integral amplitude> one and $\beta=x^2+y^2$,
$$
h(\lambda)=\frac{\pi(e^{i\lambda}-1)}{i\lambda},
$$
whose stationary contribution is $i\pi/\lambda$ and whose boundary contribution is $-i\pi e^{i\lambda}/\lambda$. Thus \b[the boxed formula is the standard nondegenerate interior stationary contribution], with the stated hypotheses; the data as printed do not determine all possible leading behaviours.
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