= Solution
Use the <Gauss collocation coefficient construction>. Find the $\nu$ <polynomial roots> $c_1,\ldots,c_\nu$ of the shifted <Legendre polynomial> $P_\nu(2c-1)$ in $(0,1)$, and form
$$
\ell_j(s)=\prod_{m\ne j}\frac{s-c_m}{c_j-c_m}.
$$
Then set
$$
\boxed{a_{ij}=\int_0^{c_i}\ell_j(s)ds,\qquad b_j=\int_0^1\ell_j(s)ds.}
$$
These are explicit <polynomial> <integrals>. For example, if $\ell_j(s)=\sum_{r=0}^{\nu-1}d_{jr}s^r$, calculate $a_{ij}=\sum_rd_{jr}c_i^{r+1}/(r+1)$ and $b_j=\sum_rd_{jr}/(r+1)$. The resulting implicit <Gauss collocation method> has order $2\nu$; its <Gaussian quadrature> nodes integrate <polynomials> through degree $2\nu-1$ exactly. The one-stage example is the <implicit midpoint rule>, with $c_1=a_{11}=1/2$ and $b_1=1$. No search over nonlinear <Runge-Kutta order conditions> systems is needed.
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