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The paper deals with a three-parameter family of special double confluentHeun equations that was introduced and studied by V.M.Buchstaber andS.I.Tertychnyi as an equivalent presentation of a model of overdamped Josephsonjunction in superconductivity. The parameters are $l,\lambda,\mu\in\mathbb R$.Buchstaber and Tertychnyi described those parameter values, for which thecorresponding equation has a polynomial solution. They have shown that for$\mu\neq0$ this happens exactly when $l\in\mathbb N$ and the parameters$(\lambda,\mu)$ lie on an algebraic curve $\Gamma_l\subset\mathbbC^2_{(\lambda,\mu)}$ called the $l$-th spectral curve and defined as zero locusof determinant of a remarkable three-diagonal $l\times l$-matrix. They studiedthe real spectral curves and obtained important results with applications tophase-lock areas in model of Josephson junction, which is a family of dynamicalsystems on 2-torus. In the present paper we prove irreducibility of complexspectral curves. We also calculate their genera for $l\leqslant20$ and presenta conjecture on general genus formula. We apply the irreducibility result tothe phase-lock areas, which are those level sets of the rotation numberfunction $\rho$ on the parameter space of the above-mentioned family ofdynamical systems that have non-empty interiors. The family of their boundariesis a countable union of analytic surfaces. We show that, unexpectedly, itscomplexification is a complex analytic subset consisting of just fourirreducible components, and we describe them. We present a MonotonicityConjecture on the evolution of the phase-lock area portraits and a partialpositive result towards its confirmation.

On spectral curves and complexified boundaries of the phase-lock areas in a model of Josephson junction