  We give an algebraic and a geometric criterion for the existence of
$G$-connections on $\mathbb{P}^{1}$ with prescribed irregular type with equal
slope at $\infty$ (isoclinic) and with regular singularity of prescribed
residue at $0$. The algebraic criterion is in terms of an irreducible module of
the rational Cherednik algebra, and the geometric criterion is in terms of
affine Springer fibers. We use these criteria to give complete solutions to the
isoclinic Deligne-Simpson problem for classical groups, and for arbitrary $G$
when the slope at $\infty$ has Coxeter number as the denominator. Among our
solutions, we classify the cohomologically rigid connections, and obtain new
cases in types $B,C$ and $F_4$.
