  We prove existence, uniqueness and regularity of solutions to the Einstein
vacuum equations taking the form $${^{(4)}g} = -dt^2 + \sum_{i,j = 1}^3
a_{ij}t^{2p_{\max\{i,j\}}}\, \mathrm{d} x^i\, \mathrm{d} x^j$$ on $(0,T]_t
\times \mathbb T^3_x$, where $a_{ij}(t,x)$ and $p_i(x)$ are regular functions
without symmetry or analyticity assumptions. These metrics are singular and
asymptotically Kasner-like as $t\to 0^+$. These solutions are expected to be
highly non-generic, and our construction can be viewed as solving a singular
initial value problem with Fuchsian-type analysis where the data are posed on
the "singular hypersurface" $\{ t = 0\}$. This is the first such result without
imposing symmetry or analyticity.
  To carry out the analysis, we study the problem in a synchronized coordinate
system. In particular, we introduce a novel way to perform (weighted) energy
estimates in such a coordinate system based on estimating the second
fundamental forms of the constant-$t$ hypersurfaces.
