  We consider a 3-parameter family of linear special double confluent Heun
equations introduced and studied by V.M.Buchstaber and S.I.Tertychnyi, which is
an equivalent presentation of a model of Josephson junction in
superconductivity. Buchstaber and Tertychnyi have shown that the set of those
complex parameters for which the Heun equation has a polynomial solution is a
union of explicit planar algebraic curves: the spectral curves indexed by
$\ell\in\mathbb N$. In his paper with I.V.Netay, the author has shown that each
spectral curve is irreducible in Heun equation parameters (consists of two
irreducible components in parameters of Josephson junction model). Netay
discovered numerically and conjectured a genus formula for spectral curves. He
reduced it to the conjecture stating that each of them is regular in $\mathbb
C^2$ with a coordinate axis deleted. Here we prove Netay's regularity and genus
conjectures. For the proof we study a 4-parameter extension of a family of
linear systems equivalent to the Heun equations. They yield an equivalent
presentation of the extension of model of Josephson junction introduced by the
author in his paper with Yu.P.Bibilo. We describe the so-called determinantal
surfaces consisting of linear systems with polynomial solutions as explicit
algebraic surfaces indexed by $\ell\in\mathbb N$. The spectral curves are their
intersections with the hyperplane of the initial Heun equation family. We prove
that each determinantal surface is regular outside appropriate hyperplane and
consists of two rational irreducible components. The proofs use Stokes
phenomena theory, holomorphic vector bundle technique, foliation of
determinantal surfaces by isomonodromic families of linear systems governed by
Painlev\'e 3 equation and transversality of the latter foliation to the initial
model.
