= Solution
Write $s_{ij}=M_{ij}^2$ and factor the three-body <Lorentz-invariant phase space>[phase space] through the pair $q=p_1+p_3$, with $s=q^2$:
$$
d\Phi_3=\frac{ds}{2\pi}\,d\Phi_2(P;q,p_2)\,d\Phi_2(q;p_1,p_3).
$$
A <Scalar particle>[scalar parent] permits integration over the <Rotation>[overall orientation]. In its rest frame let the pair have <momentum> magnitude $p$, and in the pair rest frame let particle 3 have magnitude $p^*$. After integrating the azimuth of its direction, the two-body factors give
$$
d\Phi_3=\frac{p\,p^*}{64\pi^3M_0\sqrt{s}}\,ds\,d\cos\theta^*.
$$
In that pair rest frame, particle 2 has <momentum> magnitude $M_0p/\sqrt{s}$, so $|ds_{23}/d\cos\theta^*|=2M_0pp^*/\sqrt{s}$. The <Jacobian determinant>[Jacobian] cancels both <momentum> factors, leaving
$$
\boxed{d\Phi_3=\frac{ds_{13}\,ds_{23}}{128\pi^3M_0^2},\qquad
\frac{d^2\Gamma}{ds_{13}ds_{23}}=\frac{|\mathcal M(s_{13},s_{23})|^2}{256\pi^3M_0^3}.}
$$
The physical domain is bounded by the requirement that the relative cosine lie between $-1$ and $1$. \b[The <Lorentz-invariant phase space>[phase-space] density is uniform within that domain; the event density is uniform only for a constant <Probability amplitude>[squared amplitude].] <Scalar particle>[Scalar parent] <spin> alone does not forbid resonance-dependent <Probability amplitudes>[amplitudes]. A factor $1/(s_{13}-m_R^2+im_R\Gamma_R)$ is a concrete nonuniform example. For two identical <pions>, a conventional overall $1/2!$ compensates integration over both labelings without changing the local flatness of <Lorentz-invariant phase space>[phase space].<Review of Particle Physics by Particle Data Group (2022)>[Kinematics, equations (49.22)–(49.23): three-body Dalitz density and its bounds]{ref}
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