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Let $G $ be the $\\ell$-group generated by $\\hat t_1, \\ldots,\\hat t_n$ in the free $m$-generator $\\ell$-group $\\mathcal A_m.$ We prove: (i) the problem whether $G$ is $\\ell$-isomorphic to $\\mathcal A_n$ is decidable; (ii) the problem whether $G$ is $\\ell$-isomorphic to $\\mathcal A_l$ ($l$ arbitrary) is undecidable; (iii) for $m=n$, the problem whether $\\{\\hat t_1,\\ldots,\\hat t_n\\}$ is a {\\it free} generating set is decidable. 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Let $\\hat t_1,\\ldots,\\hat t_n$ be their associated piecewise homogeneous linear functions. Let $G $ be the $\\ell$-group generated by $\\hat t_1, \\ldots,\\hat t_n$ in the free $m$-generator $\\ell$-group $\\mathcal A_m.$ We prove: (i) the problem whether $G$ is $\\ell$-isomorphic to $\\mathcal A_n$ is decidable; (ii) the problem whether $G$ is $\\ell$-isomorphic to $\\mathcal A_l$ ($l$ arbitrary) is undecidable; (iii) for $m=n$, the problem whether $\\{\\hat t_1,\\ldots,\\hat t_n\\}$ is a {\\it free} generating set is decidable. 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