= Solution
For $t>0$ the nonzero <Christoffel symbols>, apart from lower-index symmetry, are
$$
\boxed{\Gamma^0_{ii}=p_it^{2p_i-1},\qquad\Gamma^i_{0i}=\Gamma^i_{i0}=p_i/t\quad\text{(no sum)}.}
$$
Put $S=\sum p_i$ and $Q=\sum p_i^2$. Their substitution in the <Ricci tensor> gives $R_{00}=(S-Q)/t^2$ and $R_{ii}=p_i(S-1)t^{2p_i-2}$, with all off-diagonal entries zero. Thus either every $p_i=0$, or some exponent is nonzero and
$$
\boxed{S=Q=1.}
$$
The latter are the <Kasner metric> conditions; the all-zero static Minkowski solution is a separate vacuum possibility.
In an <orthonormal frame> the independent <curvature> components are proportional to $p_i(p_i-1)/t^2$ and $p_ip_j/t^2$. They all vanish precisely when the exponents are all zero or a permutation of $(1,0,0)$. The latter flat metric is Minkowski space in a Milne-type coordinate chart.
All three exponents cannot be strictly positive: in the nonstatic branch, $S^2-Q=2\sum_{i<j}p_ip_j=0$, contradicting positive pairwise products. If all three are nonzero, there is exactly one negative exponent: two negative ones would force the positive one above one, violating $Q=1$. Therefore \b[one <scale factor> contracts and two expand as $t$ increases] in that branch.
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