  The tunneling effect predicted by B.Josephson (Nobel Prize, 1973) concerns
the Josephson junction: two superconductors separated by a narrow dielectric.
It states existence of a supercurrent through it and equations governing it.
The overdamped Josephson junction is modeled by a family of differential
equations on 2-torus depending on 3 parameters: $B$ (abscissa), $A$ (ordinate),
$\omega$ (frequency). We study its rotation number $\rho(B,A;\omega)$ as a
function of $(B,A)$ with fixed $\omega$. The phase-lock areas are the level
sets $L_r:=\{\rho=r\}$ with non-empty interiors; they exist for $r\in\mathbb Z$
(Buchstaber, Karpov, Tertychnyi). Each $L_r$ is an infinite chain of domains
going vertically to infinity and separated by points called constrictions
(expect for those with $A=0$). We show that: 1) all the constrictions in $L_r$
lie in its axis $\{ B=\omega r\}$ (confirming a conjecture of Tertychnyi,
Kleptsyn, Filimonov, Schurov); 2) each constriction is positive: some its
punctured neighborhood in the vertical line lies in $\operatorname{Int}(L_r)$
(confirming another conjecture). We first prove deformability of each
constriction to another one, with arbitrarily small $\omega$, of the same
$\rho$, $\ell:=\frac B\omega$ and type (positive or not), using equivalent
description of model by linear systems of differential equations on
$\bar{\mathbb C}$ (Buchstaber, Karpov, Tertychnyi) and studying their
isomonodromic deformations described by Painlev\'e 3 equations. Then
non-existence of ghost constrictions (i.e., constrictions either with
$\rho\neq\ell$, or of non-positive type) with a given $\ell$ for small $\omega$
is proved by slow-fast methods. In Section 6 we present applications of results
and elaborated methods and open problems.
