= Solution
In the mean-conjunction approximation set $\lambda_b=\lambda_c=\Lambda$. The <resonant conjunction geometry> is then
$$
\phi_{bc}=q(\Lambda-\varpi_c)\pmod{2\pi},\qquad
\boxed{\Lambda_j=\varpi_c+\frac{\phi_{bc}+2\pi j}{q},\quad j=0,\ldots,q-1}.
$$
There are $q$ phase directions spaced by $2\pi/q$. The integer $p$ determines their temporal order: at exact resonance the <synodic period> is $P_{\rm syn}=(p+q)P_b/q=pP_c/q$, and the <conjunction> longitude advances by $2\pi p/q$ per <conjunction>. For coprime $p,q$ this visits all $q$ branches; <resonant-argument libration> broadens each direction by about its <libration amplitude of a resonant argument> divided by $q$.
\b[Mean <conjunction> and actual alignment are different for an eccentric <orbit>]. If $\Lambda$ is the true common longitude, write $x=\Lambda-\varpi_c=f_c$ and $y=M_c(f_c)$, with $M_c$ the <mean anomaly>. Because $\lambda_c=\varpi_c+y$ while the circular <planet> has $\lambda_b=\Lambda$, the exact geometric relation is
$$
\phi_{bc}=(p+q)y-px.
$$
The displayed equally spaced directions use $y=x+O(e_c)$, hence are leading-small-eccentricity geometry. With appreciable $e_c$, use <Kepler's equation> to locate true <conjunctions>; equating true and <mean longitudes> silently is not exact.
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