Solidary Solutions to Games with Restricted Cooperation

In TU–cooperative game with restricted cooperation the values of characteristic function $v(S)>0$ are defined only for $S\in \mathcal{A}$, where $\mathcal{A}$ is a collection of some nonempty coalitions of players.
We examine generalizations of both the proportional solutions of claim problem (Proportional and Weakly Proportional solutions, the Proportional Nucleolus, and the Weighted Entropy solution) and the uniform losses solution of claim problem (Uniform Losses and Weakly Uniform Losses solutions, the Nucleolus, and the Least Square solution). These generalizations are $U$–equal sacrifice solution, the $U$–nucleolus and $qU$–solutions, where $U$ and $q$ are strictly increasing continuous functions.
We introduce Solidary (Weakly Solidary) solutions, where if a total share of some coalition in $\mathcal{A}$ is less than its claim, then the total shares of all coalitions in $\mathcal{A}$ (that don't intersect this coalition) are less than their claims. The existence conditions on $\mathcal{A}$ for two versions of solidary solution are described.
In spite of the fact that the versions of the solidary solution are larger than the corresponding versions of the proportional solution, the necessary and sufficient conditions on $\mathcal{A}$ for inclusion of the $U$–nucleolus in two versions of the solidary solution coincide with conditions on $\mathcal{A}$ for inclusion of the proportional nucleolus in the corresponding versions of the proportional solution. The necessary and sufficient conditions on $\mathcal{A}$ for inclusion $qU$–solutions in two versions of the solidary solution coincide with conditions on $\mathcal{A}$ for inclusion of the Weighted Entropy solution in the corresponding versions of the proportional solution.
Moreover, necessary and sufficient conditions on $\mathcal{A}$ for coincidence the $U$–nucleolus with the $U$–equal sacrifice solution and conditions on $\mathcal{A}$ for coincidence $qU$–solutions with the $U$–equal sacrifice solution are obtained.